ATARI 8 PROGRAMMING O Concepts and Techniques 8 w 3 c 3 / ?////* / / 1 1 jjp A Data Becker book published by m ■ v#u c,n cm "‘ ob ffffffiium r* Abacus mm Software ;ii **>♦♦♦♦« 3-D Graphic programming By Uwe Braun A Data Becker Book Published by A I I lMWSIIlj) r c - Abacus iiiiiiisnl Software First Printing, Aug. 1986 Printed in U.S.A. Copyright © 1985 Data Becker GmbH Merowingerstr.30 4000 Dusseldorf, West Germany Copyright © 1985 Abacus Software, Inc. P.O.Box 7219 Grand Rapids, MI 49510 This book is copyrighted. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of Abacus Software or Data Becker, GmbH. Every effort has been made to insure complete and accurate information concerning the material presented in this book. However Abacus Software can neither guarantee nor be held legally responsible for any mistakes in printing or faulty instructions contained in this book. The authors will always appreciate receiving notice of subsequent mistakes. ATARI, 520ST, ST, TOS, ST BASIC and ST LOGO are trademarks or registered trademarks of Atari Corp. GEM is a registered trademark of Digital Research Inc. ISBN 0 - 916439 - 69-0 Table of Contents 1. Introduction 3 2. Mathematical Basis of Graphic Programming 7 2.1 Moving the Coordinate Base 10 2.1.1 Scaling the Axis 12 2.1.2 Rotation around one point 16 2.2 Plane conversion with matrix operations 19 2.3 Clipping 26 2.4 Transformations in three dimensional space 40 2.4.1 Rotation about any desired axis 36 2.5 Projection from space to a two dimensional plane 43 2.6 Perspective transformation 48 2.7 Hidden lines and hidden surfaces 56 2.8 Rembrandt and hidden surfaces 65 3. Machine language fundamentals for graphic programming 75 3.1 Speed Advantages from tables 76 3.2 Assembler routines for screen manipulation 79 3.2.1 Drawing lines 80 3.3 Operating system functions 87 3.3.1 Starting a program 88 4. Graphics using assembly language routines 107 4.1 Definition of a data structure of an object in space 108 4.1.1 Explanation of subroutines used 152 4.1.2 Description of the Subroutines of the first Main program 165 4.2 Generation techniques for creating rotating objects 169 4.1.2 New subroutines in this program 188 4.3 Hidden line algorithm for convex bodies 191 4.3.1 Explanation of the newly added subroutines 214 4.3.1.1 Errors with non-convex bodies 215 4.4 The painter algorithm 216 4.4.1 New things in the main program rotl.s 246 4.4.2 Sort algorithm 247 4.5 Entering rotation lines with the mouse 248 4.5.1 Description of the new subroutines 296 4.6 Handling several objects 298 5. Suggestions for additional development 327 5.1 Light and Shadow 329 5.2 Animated Cartoons 330 Appendices A Number systems 333 B Analytical geometry of the planes and space 335 B.l Scalar product 342 B.2 Cross product 342 C Matrix calculations 344 C.l Adding matrices 345 C.2 Multiplying matrices 346 D Bibliography 348 Index 349 L_ Abacus Software ST 3D Graphics 1. Introduction The possibilities of computer graphics are some of the most challenging reasons for working with a computer today. Dazzling computer-generated images are showing up almost everywhere--in medicine, engineering, motion pictures, music videos, television advertising, and even in newspapers like USA Today. The public is fascinated by the unlimited forms that computer graphics are taking. Some of the more sophisticated of these works are the three-dimensional, computer-animated videos used in television advertisements. One major application of computer graphics in industry is for CAD (Computer-Aided Design) systems. The integration of CAD systems into the manufacturing process is of increasing importance. Known as CAD- CAM (Computer-Aided Design - Computer-Aided Manufacturing), these systems are making significant inroads in automating many of the manufactured, assembled, and processed goods such as machine tools, automobiles, electronics, and agricultural products. Without advanced graphic data processing, the latest medical processes such as CAT scans would be difficult, if not impossible. Furthermore, three-dimensional graphic data processing has made it possible to visually represent complicated scientific relationships and to make them comprehensible (like atomic and molecular models and the DNA helix). Eventually these graphics will be integrated with advanced teaching and simulation methods, and are bound to have a profound impact on the way we think and learn. The enormous strides made in the production of integrated circuits and the increase in processing speeds of relatively new microprocessors such as the Motorola MC68000 has made it possible for the home and personal computers to enter application areas that were formerly the domain of large mainframe computers costing several hundred thousand dollars. Even now, an affordable 32-bit personal computer is just around the comer. The traditional distinctions between microcomputers, minicomputers, and mainframes are becoming increasingly blurred. Of course, even the largest mainframes are getting faster as well. The fastest computer at this time, the Cray II, has a throughput capacity of 2000 megaflops (200 million floating-point operations per second). Such high computing speeds are needed to closely simulate natural world processes with computer models. Examples are the simulation of 3 ST 3D Graphics Abacus Software ecological problems (acid rain), simulation of the human physiology, weather prediction, nuclear fusion and fission, origin of the solar system, simulation of star systems, space travel, etc. This book is intended to explore some of the possibilities of creating two- and three-dimensional computer graphics on the Atari ST computer series. To obtain a good understanding of the program sections, you should have some fundamental knowledge of MC68000 machine language. Machine language represents the lowest level of communication with the computer and contains a small number of rather simple instructions that are consequently easy to learn. For the hobbyist, knowing machine language programming makes it easier to understand the data structure of higher-level languages such as Pascal and C. However, most problems and algorithms are easier to program in a higher-level language than in machine language. For the problem of depicting and representing the 3-D wire models presented here, maximum processor speed is crucial. Machine language is clearly superior to any higher-level language in fulfilling this requirement. With these applications for the Atari ST, real-time three- dimensional graphics can be achieved. The removal of hidden lines and the shading of areas requires a considerable amount of processor time. The Cray II requires 8 minutes to create a single picture with a resolution of 2000 by 3000 pixels, with up to 30 bits of color information per picture point. In contrast, the ST manages only 640 by 400 pixels and only one bit of color information. Of course, it is possible to increase the computational capabilities of the ST with programming tricks, fast mass storage (hard disk) and large amounts of memory to solve more complex graphic problems. This book provides you with help in solving the complex programming problems of three-dimensional graphics. While the sample programs are directly tailored for the Atari ST, the techniques can be used without too much difficulty on other computers. Only the routines for hardware communication and display control (keyboard input, line drawing, surface shading (if possible) and switching between two screens) need to be tailored to another computer using an MC68000 CPU (i.e., the Apple Macintosh and Commodore’s Amiga). The subroutines for generating and handling three-dimensional graphic objects can be run on any computer with an MC68000 microprocessor. 4 Abacus Software ST 3D Graphics 2. Mathematical Basis of Graphic Programming This chapter serves as the mathematical foundation of computer¬ generated, three-dimensional graphics. As a result, the explanations are very extensive. For this reason we ask readers who are already familiar with these topics for a little patience and understanding. All computer graphic problems can ultimately be reduced to the problem of drawing points on a graphic output device (monitor screen, plotter, or printer) and to connect these points with lines. There may also be the task of shading the area delineated by the lines. For a demonstration, we will use a two-dimensional plane with one Cartesian coordinate system, familiar to everybody, whose origin lies in the lower left hand comer of the screen. Cartesian coordinate system T - 1 - 1 - 1 - 1 -r— i r r i i r» I ST display coordinate system Fig. 2.1: coordinate system and ST display coordinate system In Figure 2.1, the first problem of representing graphics becomes clear. The Cartesian coordinate system and the display coordinate system used by the ST’s software and hardware are not the same. The directions of the y-axis are opposite, and the coordinate origin is displaced. Consequently, 7 ST 3D Graphics Abacus Software an object defined in the first system is inverted in the system on the right, and is also displaced on the y-axis. At first, you might be tempted to define objects to be represented using the ST’s coordinate system. But doing this does not solve the second problem-that the display surface of every computer is limited. The ST can display only 640x400 points at its highest resolution. So, to avoid defining objects with these limitations of 640x400 points, we must be able to define an object in any desired coordinate system before displaying it on the monitor screen. In other words, we must be able to scale the object in any of the coordinate systems, i.e., change its size. All points of the defined object can then be transformed using graphics operations. This operation is called windowing. We now introduce three coordinate systems. They are: 1. world coordinate system 2. view coordinate system 3. picture coordinate system Individual objects are defined in the world coordinate system , where the calibration of the coordinate axis may be any desired unit of measurement—for example, millimeters, kilometers, years, etc. The view coordinate system accepts a portion of the world coordinate system. This is similar to an observation window in the world coordinate system. Finally, the picture coordinate system represents the physical screen display of the computer, A single point in this system corresponds to an individual pixel on the screen. This concept can be explained very simply with an example. Two objects are defined in a world coordinate system, the outlines of a house and of a church. The two outlines represent all objects that can be depicted on a plane. For example, an architect would use the outline of the house in a world coordinate system to define individual rooms and furniture. Abacus Software ST 3D Graphics Our task is to transform the observation window, together with the house that fills its surface, to the specific picture window for display on the ST’s screen. Here’s the preferred solution to the problem, using the view coordinate system: The origin of the world coordinate system is moved to the lower left comer of the observation window and scaled by a suitable factor. It now represents all points in the picture coordinate system. If the points are in the field of picture coordinates, they can be drawn and connected with lines. Cartesian coordinate system Screen system for Atan ST Fig. 2.2: Transformation of world coordinates to picture coordinates 9 ST 3D Graphics Abacus Software 2.1 Moving the coordinate base Scaling and (as we shall see later) rotation are both related to the coordinate base. To scale an object in relation to another point, or to rotate it around an arbitrary point, the coordinate origin must first be moved to the relative origin. We can illustrate this again using the house example. nx Fig. 2.1.1 One way to describe the house is to list the coordinates of the end points and to list the points which are connected with lines. For this example, the two lists are as follows: 10 Abacus Software ST 3D Graphics End point list: Point X-coordinate Y-coordinate PI 100 50 P2 100 90 P3 120 130 P4 140 90 P5 140 50 Connection list: Line from Point A to Point B LI: PI P2 L2 : P2 P3 L3: P3 P4 L4 : P4 P5 L5: P5 PI This description of a polygon, consists of a sequence of closed lines. It contains all the information necessary for representing it on the display screen. To draw the polygon, the lines’ endpoints are passed to a subroutine for drawing. As we shall see later, the polygon is also perfectly acceptable for the description of complex, three-dimensional objects. Any physical object can be closely approximated by chaining various polygons. Also, natural asymmetrical bodies such as mountains, forests, lakes and animals can be represented in a realistic manner with polygons created through fractional geometry, i.e. fractals. In addition, most man-made objects are constructed in a symmetrical manner and are easier to represent graphically. In Figure 2.1.1 the coordinate origin of the world system is moved to point P1 [ 10 0,5 0 ] . The new world coordinates (view coordinates) are obtained by subtracting the coordinates of point P1— the new origin—from the points that define the object. In general, the new world coordinates are equal to the old world coordinates, minus the coordinates of the new origin (in world coordinates). If we describe the old world coordinate axis with x and y, the new world coordinate axis with x' and y', the new origin point with NU[nx,ny3 and the point to be moved with P1 [ x, y ], we can write: Pl[xl',yl'] = Pl[xl,yl] - NU[nx,ny] 11 ST 3D Graphics Abacus Software For example, for point 5—the new origin is located at PI (100,50) = NU (100,50). The coordinates of the point to be moved P 5 (14 0,5 0) become in the new world coordinate system P5x' =140-100 = 40, P5y' =50-50 = 0. The point P5 (140,50) becomes point P5' (40,0) . This translation must be performed for every point of the object. It is possible to move the origin of the world coordinate system to any point. 2.1.1 Scaling the Axis As previously mentioned, scaling the axis refers to the coordinate origin. This can be readily seen in Figure 2.1.2. The points of the house, i.e. the X and Y coordinates, are scaled by the factor one half in the X and Y axes. The result is the halving of the length of the edges, but also a translation in the direction of the origin. If we want to avoid displacing the direction of the origin, then before scaling the origin must be moved to a point not affected by the scaling itself. The Figure 2.1.3 is an example. If we want to leave the left lower comer of the house (the point PI) in its place. The origin is moved to point PI. The picture is scaled by multiplying the X and Y values by one half and finally moving the origin to its original location. In this example this means: 1. Subtract 100 from the X-values of points P1-P5 Subtract 50 from the Y-values of points P1-P5 2. Multiply all X- and Y-values of points P1 -P 5 with the factor one half. 3. Add 100 to all X-values of points P1-P5 Add 50 to all Y-values of points P1-P5 12 Abacus Software ST 3D Graphics 13 ST 3D Graphics Abacus Software Figure 2.1.3 Scaling with factors greater than one enlarges the object. If we select different scaling factors for the X and Y axes, a distorted picture of the object results. At this point let’s briefly return to the example, at Figure 2.1, and alter the scaling factors for converting to view coordinates. With the maximum coordinates of the observation: [wxmin,wymin]; [wxmax,wymax], and the display window [vxmin,vymin]; [vxmax,vymax] one can give differing scaling factors for the two axes, Sx and Sy. In our example: Sx=(vxmax-vxmin)/(wxmax-wxmin) Sy=(vymax-vymin)/(wymax-wymin) 14 Abacus Software ST 3D Graphics Before scaling, the origin of the world system is moved to the left lower comer of the observation window [wxmin, wymin], since this point is the data point of the scaling. The result of the conversions is therefore: 1. Move the origin to the point W1 [wxmin,wymin] by subtracting wxmin from all of the X coordinates and wymin from all of the Y coordinates. 2. Multiply all X and Y values of the points with the factor j Sx. If the relationship of height to width is equal for hpth c <»•/*'' ‘ oa «UL windows, then Sx=Sy. A*}*^ xV r °° 3. Convert to the display system by multiplying the Y values by -I and adding of the maximum Y value to these Y values (for the monochrome ST this is 399). This moves the origin to the upper left comer of the screen. The third step of converting the Y values to the screen display of the ST is always the same. During the description we shall limit ourselves to the view system. If during subsequent discussions no special reference is made to this step, you should remember that if it is not performed, all objects appear inverted on the screen after the drawing is completed. The location of the picture window in the view system is not fixed to the origin, but is movable in the total view system. However, the three conversions must be followed by another conversion-moving the window to point VI [vxmin,vymin] . Basically the conversion of an object is the opposite of the conversion of a coordinate system. Therefore, when moving the picture window and the object to the point VI [vxmin, vymin] , the coordinates of this point (vxmin and vymin) must be added to all object coordinates. Summarizing the conversion of the world system into the view system: 1. Move the origin to the point W1 [wxmin,wymin] by subtracting wxmin from all X coordinates, and wymin from all Y coordinates. 2. Multiply all X values of the points by the factor Sx=(vxmax-vxmin)/(wxmax-wxmin) , the Y values with the factor Sy= (vymax-vymin) / (wymax- wymin) . 15 ST 3D Graphics Abacus Software 3. Move the window and the object to the point VI [vxmin, vymin] by adding vxmin to all X values, and vymin to all Y values. 4. Convert to the display system by multiplying the Y values by -1 and adding the maximum Y-value to these Y values (for the highest resolution this value is always 399). 2.1.2 Rotation around one point The rotation of an object is related to a single point, just as we found out in the previous section on scaling. To start the conversion, a single point is rotated around the origin. Since the rotation occurs around the single origin point, the data point of the rotation angle is the connecting line between coordinate source and the point to be rotated. See Figure 2.1.4. The point PI (xl,yl) is moved by rotation around the angle p of the origin to the point P2 (x2,y2). We must define the sign of the angles a and p as + or Following the conventions of mathematics, we designate 16 Abacus Software ST 3D Graphics the angles as positive when the rotation moves the positive X axis to the positive Y axis. Expressed differently, positive angles are measured in the counterclockwise direction. For the angle between the connecting line from 0,0 to P1 and the X-axis, the relationships are: 1) SIN(alpha) = Yl/C 2) COS(alpha) = Xl/C 3) SIN (alpha+beta)=Y2/C 4) COS(alpha+beta)=X2/C with C=V (X1 2 +Y1 2 ) W (X2 5 +Y2 2 ) . The addition theorems for the angle functions SIN and COS are as follow (we won’t derive them here): 5) SIN(Alpha+Beta)=SIN(Alpha)*COS (Beta)+COS(Alpha)*SIN(Beta) 6) COS(Alpha+Beta)=COS(Alpha)*COS(Beta)- SIN(Alpha)*SIN(Beta) By combining these equations, X2 and Y2 can be calculated quite easily: 7) X2/C=COS(Alpha)*C0S(Beta)- SIN(Alpha)*SIN(Beta) gives us 8) X2= COS(Alpha)*C* COS (Beta) - SIN(Alpha)*C* SIN(Beta) from 1) follows 9) X2=Xl*COS(Beta)-Y1*SIN(Beta) 10) Y2=Yl*COS(Beta)+X1*SIN (Beta) 17 ST 3D Graphics Abacus Software As an example of rotation, we will rotate the house in Figure 2.1.5 by an angle of 30 degrees around the origin. The points P1-P5 become points R1-R5, as can be seen on the example at Point PI. R1X=P1X*C0S (30)-P1Y*SIN(30) R1Y=P1Y*COS(30)+P1X*SIN (30) From PI (100,50) follows R1 (61. 6,93.3). According to the same principle, the remaining points are likewise converted. Figure 2.1.5 18 Abacus Software ST 3D Graphics 2.2 Plane conversion with matrix operations After learning about the conversions, translations, scaling and rotations described in the previous chapter, we are now able to draw on the screen any object previously defined in a two dimensional coordinate system, in any selected size and viewing angle. One drawback to this method is that several arithmetic operations are required for each and every point of the object. Right now we’ll combine these conversion operations into a single matrix operation. (Explanations of matrix operations are found in the Appendix). Therefore it becomes possible to apply the conversions to the array and then to multiply the resulting array with every point of the object. To make the array operations usable for the point coordinates of the plane, the point coordinates are first converted to array form. There are basically two ways to convert these: with column vectors (2,1), or with line vector (1,2) arrays. A conversion array (2,2) is used to multiply a line vector with the transformation array, where the transformation array must be multiplied with the column vector, (number of columns A = number of rows B). In this book we shall write the point coordinates as line vectors P and the multiply this line vector with the transformation array. This sequence of multiplication simplifies, purely subjectively, the creation of the transformation matrices. If you multiply a line vector (1,2) with a quadratic array (2,2), you will obtain as a result another line vector (1,2), which represents point coordinates. The individual point operations can be expressed by a suitable transformation matrix T. For scaling the X axis by the factor 2, the array S1 is valid. It is also possible to quadruple the Y values using transformation array S2. The two scaling steps can be by multiplying SI and S2 with array S3. Si 2 0 0 1 S2 = 1 0 0 4 S 3 = S 1 * S 2 = 2 0 0 1 ★ 1 0 0 4 s 3 0 4 19 ST 3D Graphics Abacus Software For rotation, R x is valid for one counter clockwise rotation; from trigonometry, a clockwise rotation occurs with r 2 . From Figure 2.1.5, the movement of point P1 [ x 1, y 1 ] to point P2[x2,y2], results from multiplying P1 with R. Rl = cos(b) -sin(b) sin(b) cos(b) R -2 = cos(-b) sin(-b) = cos(b) -sin(b) -sin {-b cos(-b) sin(b) cos(b) P2[X2,Y2] = [XI, Yl] * cos (30) sin(30) -sin(30) cos (30) Several rotations in succession can be carried out by multiplying the rotation matrices. Unfortunately, this array form does not permit translation (origin relocation). For this you can add a dimension to the vectors. Every n-dimensional object can be represented in a (n+1) space in innumerable many ways. In a three dimensional space there are infinite possibilities for laying out the X-Y plane we have just observed. The additional dimension is known as Z coordinate of the X-Y plane. For two dimensional objects, its value is always one. The X and Y coordinates remain unchanged: the line vector [x,y] becomes the line vector [x,y,l]. The array for the translation of the source at point D is as follows: 110 T = 0 10 -DX -DY 1 Every point of the object must be multiplied with this array to move the origin of the world coordinate system to the point (DX, DY). For the point P [ x, y, 1 ] the result is: new point in world coordinates p' =p*T 10 0 P'[x',y',l] = [x,y,1] *010= [x-dx,y-dy,1] -DX -DY 1 20 Abacus Software ST 3D Graphics You can combine two displacements by using array multiplications. First the origin is moved to the point [DX,DY, 1 ] and then to the point [AX, AY, 1 ] of the new coordinate system. The two translation matrices Tl and T2 are as follows: 10 0 10 0 T 1 =010 T 2 =010 DX -DY 1 -AX -AY 1 Multiplication of the matrices results in T 3 : 10 0 10 0 t 3 =t 1 *t 2 = 0 1 0 * 0 1 0 -DX -DY 1 -AX -AY 1 10 0 T 3 = 0 1 0 -DX-AX -DY-AY 1 P'[x',y',l]= P[x,y,1]*T 3 = [x-DX-AX,Y-DY-AY,1] The scaling array S can be defined in the new system: SX 0 0 S = 0 SY 0 and P' = P * S 0 0 1 and finally the rotation array R cos (a) sin (a) 0 R(a) = -sin(a) cos (a) 0 0 0 1 21 ST 3D Graphics Abacus Software Scaling as well as rotation, viewed individually, may be carried out in a series through array multiplications. The array multiplication is normally not commutative, i.e. Ti * T 2 is not necessarily identical with T 2 * Ti. However, the multiplication of the following array types is commutative: 1) Translation * Translation 2) Scaling * Scaling 3) Rotation * Rotation around the same axis 4) Scaling •k Rotating Type 4 (scaling and rotating) is only valid when both scale factors (Sx, Sy) are identical. These fundamentals enable us, through a combination of several array operations, to rotate an object around a selected point V[vx,vy,l] using a series of several array operations. The various operations are: 1. Shifting the origin to point V 2. Rotation around point V by an angle of alpha 3. Shifting of the origin to the original point Three matrices T x , R x and T 2 are required: l 0 0 co3(a) sin(a) 0 Ti - 0 1 0 Rl = -sin (a) cos(a) 0 -vx -vy 1 0 0 1 1 0 0 t 2 = 0 1 0 vx vy 1 For the multiplication array M 1# the result is: Mi = T x * Ri * T 2 and for every point follows: P' = P * Mi 22 Abacus Software ST 3D Graphics The sequence of matrices is decisive in these operations and must occur from left to right. It is possible however, to first calculate intermediate results, but these must be used in the "right" sequence. In this example, there are two possible ways to proceed: 1. First calculate from Z i =T 1 *R 1 and then M 1 =Z 1 *T 2 2. First calculate from Z 2 =Ri*T 2 and then M 2 =T 1 *Z 2 The first case is explained in detail. Z 1 =T 1 *R 1 : 0 0 cos(a) sin(a) 0 1 0 * -sin(a) cos (a) 0 -vy 1 0 0 1 cos (a) sin(a) 0 -sin(a) cos(a) 0 cos(a)+vy*sin(a) -vx*sin(a)-vy*cos(a) 1 and now M i =z 1 *T 2 : cos(a) sin(a) M x = -sin(a) cos(a) -vx*cos(a)+vy*sin(a) -vx*sin(a)-vy*cos(a) 0 0 * 1 10 0 0 1 0 = vx vy 1 sin(a) 0 cos(a) 0 vx*sin(a)-vy*cos(a)+vy 1 cos(a) -sin(a) -vx*cos(a)+vy*sin(a)+vx 23 ST 3D Graphics Abacus Software If point PI [x, y, 1 ] is multiplied with this array, the result is point PI' [x' , y' , 1], the point PI which was rotated around the angle alpha at point VI [vx, vy, 1]. This connection can be recognized in Figure 2.2.1 and should be performed as example for point PI. PI [x, y, 1 ] * Ml = cos (a) sin (a) 0 [x,y,l] * -sin (a) cos (a) 0 —vx*cos(a)+vy*sin(a)+vx -vx*sin(a)-vy*cos(a)+vy 1 Pl[x,y,z]= f[x*cos(a)-y*sin(a) -vx*cos (a) +vy*sin (a) +vx] , [x*sin(a)+y*cos(a)-vx*sin(a)-vy*cos(a) +vy], [1] ] You can see that when the rotation point and the point to be rotated are identical, therefore x=vx and y=vy, the expression for the line vector of the point at [ vx, vy, 1 ] = [ x, y, 1 ] degenerates. That means that the point coordinates do not change. 24 Abacus Software ST 3D Graphics The house already familiar in Figure 2.2.1 shall be rotated by the angle alpha=30 degrees around the point VI [ vx, vy, 1 ] = [ 12 0,8 0, 1 ]. As an example this is carried out on point P2[100,90,l]. P2x'=100*cos(30)-90*sin(30)-120*cos(30)+80*sin(a)+120 P2y'=100*sin(30)+90*cos(30)-120*sin(30)-80*cos (30)+80 P2'=[97.68,78.66,l] and finally for the remaining points P1-P5. PI'=[117.68,44.02,1] P2'=[97.68,78.66,1] P3'=[95,123.30,1] P4'=[132,32,98,66,l] P5'=[143.66,59.02,1] This procedure also permits you to change the point for scaling to any location in the coordinate system. In the following, you can see the buildup of the transformation array. First the coordinate origin is moved to point K1 [ kx,ky,1] with translation array T 1? then scaling with array Si, using scaling factor Sx and Sy, and finally moving the origin to its original location using translation array T 2 . For every single point this means p' [x' ,y', 1] = p [x,y, 1] *T 1 *S 1 *T 2 . -kx 0 0 1 0 -ky 1 Sx Si- 0 0 0 0 Sy 0 0 1 1 t 2 = 0 kx 0 0 1 0 ky 1 Sx 0 0 T-l* Si* T 2 = 0 Sy 0 kx*(l-Sx) ky*(1-Sy) 1 P'[x,y,1] =P'[x*Sx+kx(1-Sx),y*Sy+ky(1-Sy),1] In this example Sx=Sy=0.5. 25 ST 3D Graphics Abacus Software 2.3 Clipping As we transformed the object coordinates to the display coordinate system, we assumed that all points in the object can be represented in the picture coordinate system. When we define a window in the world system, some objects may be completely pushed out of the view of the window, or objects are cut in half by the window. This means that one or several connecting lines of the points cut the comers of the observation window. Figure 2.3.1 To avoid these incomplete objects, we can test the coordinates to make sure they lie within the borders of the window. This method slows down the drawing procedures considerably. Therefore it is better to determine before drawing a line if the line is completely visible, partially visible, or not visible at all. The window is surrounded by eight equally large surfaces to determine the exact position of the line to the window. Now the exact location of a line can be determined by comparing its 26 Abacus Software ST 3D Graphics coordinates to the window borders. A code containing four bits can be used to represent the relative position of a line outside of the window. Bit number 3 210 Figure 2.3.2:CIip-Window In the Figure 2.3.2 the position of a point outside a window is repeated by a set bit as follows: bit position 0 = Point is left of the window 1 = Point is right of the window 2 = Point is below the window 3 = Point is above the window 27 ST 3D Graphics Abacus Software The code [0,1,0,1] means the following: the point is to the left and below the window. With this information, it is possible to calculate the points of intersection of the lines with the window edges by including them in the equation. This leads to a quadratic equation system whose solution requires several multiplications and divisions. For our purposes, we want to limit the number of multiplications and to replace them when possible with other mathematical operations. We do this for two reasons. The first is for speed since multiplication requires about eight to ten times the calculation time of addition. The second is the fact that the result of multiplication, with the same number of significant positions of the operands, has a larger relative error. To get an optimal solution of the line-clipping problem requires a programming language which permits bit manipulation. This was developed by Cohen and Sutherland. Since the efficiency of the Cohen- Sutherland clipping algorithm is so great, it is sometimes implemented in the hardware of some graphic terminals. The starting point of the algorithm is to divide the plane into the nine areas previously illustrated. For every line which is to be "clipped”, you must determine a center point and on the basis of its position relative to the window. The calculation of the center point of a line AB is simple. Just add the X and Y coordinates of the end points and divide them by two. Mx= (ax+bx) /2, My= (ay+by) /2. Division by two is performed by microcomputers easily by a single right shift and this explains the speed of the algorithm. The 8 different positions of the end points relative to the window are illustrated in Figure 2.3.2. Before calling the clip-routine, you must first test to see if the two end points are visible. If any of the bits are set, then some portion of the line is not visible. In Figure 2.3.2 both A and B are above the upper window edge, and therefore the line AB is not visible and no longer needs to be considered. You can calculate the position of the points by "ANDing" their codes and then testing for a "not zero” condition. For lines which have no common position parameter, for example the line CD, positions are determined with two separate procedures. First the right and then the left intersecting points with the clip-window. 28 Abacus Software ST 3D Graphics First calculate the midpoint Ml of line CD. After determining the position code of the point Ml, it is compared with the code of the right endpoint D. If a single bit of these codes is the same, then the partial line MID does not have to be considered further, and the right endpoint D is replaced with the point Ml which was just determined. Now the midpoint of line CM1 (M2) , is calculated and tested again with the right endpoint, this time Ml. If both points are not on one side, M2 becomes the new left endpoint and the right endpoint remains Ml. Next search the midpoint of the line M2M1. This procedure is continued until a new calculated midpoint is equal to one of the two end points used for calculation. After completing the algorithm, the last left endpoint is the desired intersecting point with the window. The intersecting point is stored and the two starting points C and D are interchanged. With the same procedure the intersection with the left window edge is determined. At the start of the routine, if you find that an endpoint is already inside the window, this endpoint must be stored. The line ST causes a problem. The two end points S and T are not on the same window side and the line does not intersect the window. A comparison of the first center point T1 shows it matching both end points. The points T1 and T are both to the right of the window and point S below the window. You can thus define a new ending criteria--# a new midpoint lies outside of the window and matches both end points of the line, then the line is not visible. 29 ST 3D Graphics Abacus Software 2.4 Transformations in three dimensional space A small warning before we start: Thinking in three dimensions requires a period of adjustment for the non-mathematically oriented reader. It may be necessary to read this chapter several times before the concepts can be fully understood. Starting with the two dimensional X-Y-coordinate system, there are two ways to introduce a right angle coordinate system to describe three dimensional space. They are the right-hand and the left-hand coordinate system which differ only in the orientation of the negative Z axis. A coordinate system is called a right-hand coordinate system when a screw with a right-handed thread (a normal wood screw) moves in the direction of the positive Z axis when it is turned from the positive X axis in the direction of the positive Y axis. See Figure 2.4.2. The right-hand coordinate system is used extensively in mathematics while some computer graphic books select the left-hand coordinate system. 30 Abacus Software ST 3D Graphics Mathematical problems can be solved in either system and one system can easily be turned into the other. We shall use both systems. The transformations in three dimensional space will be explained on a right- hand coordinate system, the perspective transformations on a left-hand coordinate system. z z Right-hand Left-hand Figure 2.4.2 All operations in a two dimensional space are special cases of corresponding operations in three dimensional space. In the extended coordinate system, the line vector of a point is expressed as: P [x,y, z, 1]. To move the origin to the point V[vx,vy, vz, 1], use the matrix T1: T i = 1 0 0 •vx 0 1 0 ■vy 0 0 1 -vz 0 0 0 1 So for every point: [x,y,z,l] * Ti = [x-vx,y-vy,z-vz,1] 31 ST 3D Graphics Abacus Software The scaling matrix is similar. A scaling factor for the Z axis (Sz) is added: Sx 0 0 0 S 1 = 0 Sy 0 0 0 0 Sz 0 0 0 0 1 For every point: [x, y, z, 1] * = [x*Sx, y*Sy, z*Sz, 1 ] Rotation is limited to the three rotation axis: X,Y, and Z. We are already familiar with rotation about the Z axis from the earlier 2D description. The 3D description is derived by assuming that the positive Z axis projects from the drawing surface. The coordinates of the axis about which rotation takes place, does not change, in this case the Z coordinates retain their values. cos(zw) sin(zw) 0 0 R z = -sin(zw) cos (zw) 0 0 0 0 10 0 0 0 1 32 Abacus Software ST 3D Graphics Figure 2.4.3 We must also allow for setting a positive turning angle for the rotation about the X and Y axes. A solution which can be applied to both the left- hand and right-hand coordinate systems uses the following definitions: Rotation axis positive angles are measured from Z-axis X- to Y-axis Y-axis Z- to X-axis X-axis Y- to Z-axis From this follow the matrices for rotation around the X and Y axis R x and Ry. 33 ST 3D Graphics Abacus Software 10 0 0 R x = 0 cos(xw) sin(xw) 0 0 -sin(xw) cos(xw) 0 0 0 0 1 cos(yw) 0 -sin(yw) 0 R y = 0 10 0 sin(yw) 0 cos(yw) 0 0 0 0 1 For the coordinate system this means that if you look from a positive axis in the direction of the coordinate origin, a positive angle describes a counterclockwise rotation. In a left-hand coordinate system a positive angle describes a rotation in the clockwise direction. This definition applies to a fixed coordinate system in which the objects are rotated. The other type of representation would be the fixed placement of the object and the rotation of the coordinate system. The two types differ only in the sign of the rotation angles. This means that if the object is rotated about the angle alpha, or the coordinate system is rotated about angle alpha, the result in both cases will be the same. In three dimensional space the point of the rotation, as in the two dimensional plane, is the origin. If you want to rotate an object around another point, it is first necessary to move the origin to that point. The required steps are: 1. Change the origin to the point B[bx,by,bz, 1] using translation matrix Ti. 2. Rotate around the Z axis with rotation matrix R x . 3. Retranslate the origin using translation matrix T 2 10 00 cos(a) -sin(a) 0 0 T]_= 0 1 0 0 Rl= -sin (a) cos (a) 0 0 0010 0 0 10 -bx -by -bz 1 0 0 01 10 0 0 T 2 = 0 1 0 0 0 0 10 bx by bz 1 34 Abacus Software ST 3D Graphics Figure 2.4.4 Let’s assume that you want to rotate an object about around all three axes. It is then possible to combine the rotation matrices R x , Ry and R z by multiplying with Rg. In contrast with the combination of rotations about the same axis in this example the sequence of multiplications is important, i.e. R x *Ry*Rz yields a result different from R z *Ry*R x . A point with a positive Z value is rotated 90 degrees around both the Z and X axes. If the rotation is first made about the Z axis, the coordinates do not change, X- and Y-coordinates are equal to zero, and the subsequent rotation about the X axis rotates the point to the Z=0 level; which is the X-Y plane. If the first rotation is about the X axis, the point is transferred to the Z=0 level and the subsequent rotation about the Z axis rotates the point into the Y=0 level, which is the level between the X and Z axes. This example shows why it is necessary to follow the sequence of rotations during program generation. 35 ST 3D Graphics Abacus Software 2.4.1 Rotation about any desired axis Up to now we have only considered rotation about one of the coordinate axes; with suitable combinations of various transformations we can turn an object around any desired line in space. Two points Pl[xl,yl,zl] and P2 [x2,y2, z2] are sufficient to describe a point in space. The equation through these two points: x = xl + t*(x2-xl) y = yl + t*(y2-yl) with t elements from R z = zl + t*(z2-zl) 36 Abacus Software ST 3D Graphics Since the problem for rotation about one coordinate axis has already been solved, we want to transform a rotation axis in such a way that it will coincide with the negative Z axis. The sequence of the transformation looks like this: Displacement of the coordinate origin to the point PI [xl, yl, zl] on the line. Rotation about the angle xw on the X axis, so that the rotation axis lies in the X-Z plane. Rotation of the angle y w about the Y-axis until the rotation axis coincides with the negative Z axis. It is now possible to rotate the desired angle zw about the Z axis since it matches the rotation axis. If one looks from PI to P2 a positive angle will rotate an object in a counterclockwise direction. 37 ST 3D Graphics Abacus Software To transform back to the original we need: Rotation of the angle -yw around the Y axis Rotation of the angle -xw around the X axis Displacement of the coordinate origin at the starting point. The only problem is the determination of the angles xw, yw, which can be derived from the equation. As in Figure 2.4.7 we imagine that the coordinate origin is already moved to point P1. Then the coordinates of the point P2' [x2-xl,y2-yl, z2-zl] represent the direction vector of the lines. This vector is now projected on the Y-Z plane, whereby the term projection should be taken literally. In addition you should imagine the vector G [gx, gy, gz] = G [x2-xl, y2-yl, z2-zl] illuminated by light rays, parallel to the X axis and originating from the positive X axis. The shadow created in the Y-Z plane is the vector L [0, gy, gz] and the angle alpha between vector L and the positive axis Z is the desired angle xw. In a rotation about the X axis, a positive angle describes the rotation of a point from the positive Y axis in the direction of the positive Z axis. The angle alpha is positive and the rotation matrix is as follows: R x 10 0 0 cos(a) sin (a) 0 -sin(a) cos (a) 0 0 0 0 0 0 1 38 Abacus Software ST 3D Graphics sin(a) = gy/1 and cos (a) = gz/1 For the rotation matrix R x this means: 1 0 0 0 r x — 0 gz/l gy/i 0 0 -gy/l gz/1 0 0 0 0 1 After this transformation, the vector G (P IP 2) lies in the plane located between the positive Y and positive X axis. The angle gamma, which we defined to be positive, is the desired angle (yw), which rotates the vector G with one rotation about the Y axis on the negative Z axis. The rotation matrix Ry: 39 ST 3D Graphics Abacus Software cos (g) 0 0 1 sin(g) 0 0 0 -sin(g) 0 0 0 cos (g) 0 0 1 It is possible to divide the angle gamma into the partial angles beta and the right angle alpha’ (90 degrees), between the positive X and negative Z axes. Through rotation about the X axis the X coordinate of the point P2 has not changed, whereas the Y c oordinate has beco me zero. The sum of the vecto r G [gx, gy, gz ] g =V~ (gx 2 +gy 2 +gz ^) is therefo re identical to g =V" (gx^+z l z ) . From this follows z ' =V” [g z -gx 2 ) and from 1 =V (gy 2 +gz 2 ) = V (g 2 -gx 2 ) results in z ' : z' = 1. For the angle beta the following relationships result: sin(b) = 1/g and cos (b) = gx/g The rotation angle gamma is composed of beta plus 90 degrees, ga - b + 90 40 Abacus Software ST 3D Graphics From tlffe addition theorems for sine and cosine we get: sin <ga)=sin(b+90)=sin(b)*cos(90))+sin(90)*cos(b) sin(ga) = sin (b+90) = cos(b) cos(ga)=cos(b+90)=cos(b)*cos(90)-sin(90)*sin(b) cos (ga) = cos (b+90) = -sin(b) Since the rotation angle is measured positive, it is possible to include the information just acquired directly into the rotation matrix. -sin(b) 0 0 1 cos (b) 0 0 0 -cos(b) 0 0 0 -sin(b) 0 0 1 with the references to the angle functions: -1/g 0 -gx/g 0 0 10 0 gx/g 0 -1/g 0 0 0 0 1 After these preparatory transformations, the rotation takes place about the desired angle za about the rotation axis, which is the connecting line between PI to P2. The matrix for this is: cos(zw) sin(zw) 0 0 Rz =- sin(zw) cos(zw) 0 0 0 0 10 0 0 0 1 41 ST 3D Graphics Abacus Software The inverse transformation matrices: The transformations for one point -1/g 0 gx/g 0 Ry -1 =0 1 0 0 -gx/g 0 -1/g 0 0 0 0 1 10 0 0 R x 1 = 0 gz/1 -gy/1 0 0 gy/l gz/1 0 0 0 0 1 10 0 0 T -1 =0100 0 0 10 xl yl zl 1 P'[x',y', z',l]=[x,y, z, 1]*T*R x *R y *R z *R y -1 *R x -1 *T~ 1 In these cases the rotation matrices R x etc. are combined through multiplication. The translations are performed separately. 42 Abacus Software ST 3D Graphics 2.5 Projections from space to a two dimensional plane A window can be made for observation in 3D space just as it can on a 2-dimensional plane. The position of the window and its orientation relative to the world system is purely arbitrary. For definition of this observation window you should imagine a second coordinate system, the view system inside the world system. Its origin lies in the left comer of the observation window. As a position parameter which describes the position of the view system relative to the world system, the two points ORP (Observation reference point) and ODP (Observation direction point) are sufficient, both of which are defined in the world coordinate system, as well as perhaps an inclination angle between positive Y and positive V axis (za), which describes a rotation of the U-V plane about the Z axis. The view system, as illustrated in Figure 2.5.1 is a left system. The orientation of the positive Z axis is opposite to the world coordinate system. 43 ST 3D Graphics Abacus Software For clarification: Every scene defined in the world coordinate system, such as an airport for a flight simulator, can be viewed from any point inside this scene. The only parameters required are the observation reference point (ORP), which in comparison with a camera, would represent the film, and the observation direction point (ODP), which determines the direction in which the observer (the camera) is looking. The additional angle used (za) between positive Y and positive V axes describes a rotation of the camera about the longitudinal axis of the objective. The focal point of the lens at which all light rays passing the objective meet, would in this example be on the negative Z axis. Keeping to the example of the camera, exposing a picture must transform the entire scene into the view system (U-V-Z’). This transformation, which appears complicated at first glance, has already been solved: it is the rotation about an arbitrary axis. The points PI and P2 of the axis of rotation are replaced by the points ORP and ODP and the angle za describes the inclination of the V axis to the Y axis. All operations relate to the observation reference point (ORP [orx, ory, orz] ), the positive axis of the observation coordinate system (view-system) points to the observation direction point (ODP [odx, ody, odz]). Both points are described in world coordinates and the rotation matrix rotates the vector G[odx-orx, ody-ory, odz- orz] to the negative Z axis of the world coordinate system. After fitting the V axis, the object, which was subjected to the same operations, is available in the view coordinates. Not quite, though, since the two coordinate systems still differ in the orientation of the Z axis. Therefore after fitting the V axis, all Z values must be multiplied by the factor -1 which corrects the orientation of the Z axis. The last step is a mathematical cosmetic which is required only because of the starting model of the positive Z axis of the left-hand coordinate system. If one views the result of the transformation as a right-hand system, the last step can be omitted. Let us combine the steps again, considering the steps necessary for rotation around any desired axis. 44 ST 3D Graphics Abacus Software 1. Shifting the origin to the observation reference point ORP via the translation matrix Tj. 1 0 0 -orx 0 0 0 1 0 0 0 10 -ory -orz 1 2. Rotation around the X axis until the vector G[odx- orx, ody-ory, odz-orz] = [gx / gy,gz] lies in the Y-Z-plane. 10 0 R x = 0 gz/i gy/i 0 -gy/i gz/i 0 0 0 with 1 = V(gy2+gz 2 ) 0 0 0 1 3. Rotation about the Y axis until the vector G[gx, 0, z'] meets with the Z axis: -1/g 0 -gx/g 0 0 10 0 gx/g 0 -1/g 0 0 0 0 1 with g = V (gx 2 +gy 2 +gz 2 ) l = V7gy 2 +gz 2 ) z' = 1 4. Rotation of the Z axis around the za angle for adaptation of the inclination of the V axis: R z cos(zw) -sin(zw) 0 0 sin(zw) 0 0 cos(zw) 0 0 0 10 0 0 1 46 Abacus Software ST 3D Graphics 5. Multiplication of the Z coordinates with -1 to convert from the right-hand to the left-hand coordinate system. Mi 10 0 0 0 10 0 0 0-10 0 0 0 1 The object now lies in the left-hand coordinate system U-V-Z’ and can be projected on the display, the plane suspended between the U and V axis via a suitable perspective transformation. 47 ST 3D Graphics Abacus Software 2.6 Perspective transformation Since the representation of objects on the screen is limited to two dimensions, we have to simulate the third dimension, the Z coordinate, in the two-dimensional plane. The method we used, the central projection, defines a point in space (the focal point of a lens) at which visual rays emanating from the object meet. The size of the objects represented on the display screen is directly proportional to their distance from this focal point. Equal size objects which are farther away are shown smaller than objects which are closer to the observer. T l i *z Figure 2.6.1: Perspective The coordinate system from Figure 2.6.1 is, as already indicated, another coordinate system and the plane suspended between the positive U and the positive V axis at point z’=0 will represent the screen. The center of the projection (focal point) is located on the negative Z axis at point PROZ[prozx, prozy, prozz ' ] = [0, 0, prozz' ]. The position of the point tc be viewed P[pu,pv,pz], appears to be located behind the 48 Abacus Software ST 3D Graphics observation plane. The line through these two points is described by the following equation: u = plu + (prozu-plu)*t v = plv + (prozv-plv)*t z' = plz' + (prozz' -plz') *t = 0 , the plane lies at z'=0 => t “ -plz'/(prozz'-plz') u = plu - (prou-plu)*plz'/(prozz'-plz') v = plv - (prozv-plv)*plz'/(prozz'-plz') z' =0 with prozu=prozv=0: u = plu + plu*plz'/(prozz'-plz') v = plv + plv*plz'/(prozz'-plz') z = 0 Since prozz' is negative and plz' is positive, the denominator (prozz' -plz') becomes negative, and with larger distances between focal point PROZ and point PI, the point coordinates (in the projection plane) plu' or plv' become smaller. We are now in the position to project a three-dimensional representation of the object on the screen and the distance of the projection-center object is comparable to the focal length of a camera lens. A short length corresponds to a wide-angle lens and a larger distance to a telephoto lens. The projections described are valid for the special case of the projection plane at the point z' = 0. The project plane can be moved freely on the z’ axis and can be behind the object or also behind the eye. 49 ST 3D Graphics Abacus Software In this illustration the projection center is at the point PROZ, while the object to be projected is the connecting line between the points PI P2. d designates the location of the projection plane on the Z’-axis, which can be moved arbitrarily in either direction. If the projection center and projection plane (d=PROZ) match, all objects degenerate to a single point, the center of the projection. The size of the projection can be changed by moving the projection plane. For the line between projection center PROZ and object point P1 the two point equation holds: u = plu + (prozu-plu)*t v = plv + (prozv-plv)*t z' = plz' + (prozz'-plz')*t = d 50 Abacus Software ST 3D Graphics The Z’-coordinate of the projection plane is d, and from the equation for the Z’-coordinate it follows: t = (d-plz' ) / (prozz' -plz' ) inserted into the linear equation results in the projection coordinates: u' = plx + [(prozu-plu) * (d-plz')] / (prozz'-plz') v' = p1v + [ (prozv-plv) * (d-plz')] / (prozz'-plz') z' = d Every point P[u,v,z',l] is transformed into the display coordinates P ' [u',v',d,l]. The coordinates u' and v' represent a point on the screen. The equation derived from Figure 2.6.1 comes from the special case where the projection center lies on the Z axis prozu=prozv=0 and when the projection plane is on the z'=0 plane, d=0. The following illustrations show how the selection of the various observation parameters (ORP, PROZ, d) influence the appearance of the projection. The coordinate origin of the display is in the lower left comer of the screen. 51 ST 3D Graphics Abacus Software • projection center Figure 2.6.3 52 Abacus Software + Z ST 3D Graphics resulting projection Figure 2.6.5 54 Figure 2.6.6 ST 3D Graphics Abacus Software 2.7 Hidden lines and hidden surfaces Up to now we have been in the position to project wire models of objects on the screen. The action sequence of most any computer animation is set up with the help of 3-D wire models. Wire models can be handled in real¬ time and thus shorten the development of the animation sequence considerably. Once the sequence is set, the computer calculates the visible surface and color nuances and light reflections of the objects for every intermediate point of the movement, according to the illumination. Generally the scan line algorithm is used. Seen from the eye, the vision rays are tracked through each pixel of the display (= projection plane) to the individual objects. The visual ray is either reflected, absorbed, or wholly or partially transmitted by various objects with differing surface characteristics. Under certain conditions the visual ray splits, such as on a glass surface, into a reflected and a second visual ray which passes through the object, naturally both must be tracked. This explains the computation time of about 10 minutes which even super-computers like the Cray II require for a picture. Since by conservative estimate the throughput of the Cray II is superior to that of the Atari ST by a factor of about 10,000 to 15,000, it should be clear that the ST is somewhat “under powered" for such calculations. Therefore we will limit ourselves to the “surface algorithms" and will not determine the visibility of every point, but just for each surface of the object. These algorithms are fast. To be accurate, they are valid only tor convex bodies, and in the version presented here the surfaces of the bodies must also be convex. 56 Abacus Software ST 3D Graphics Figure 2.7.1: Convex and Concave Surfaces With convex polygons the line connecting two points on the polygon lies within the polygon, whereas in convex bodies the connecting line between two points on the surface passes through the body or runs along the surface. Formulated differently, convex polygons have at least one inner angle which is larger than 180 degrees. For these surface algorithms we must expand the object definition, which up to now consisted of the point and line list, to include a surface list. The surface list contains a description of each surface by the lines which border the surface. 57 ST 3D Graphics Abacus Software The two surfaces I and II would be described in the surface list as follows: Surface Line from point to point I Pl,P4 P4,P3 P3,P2 P2, Pi II P5,P6 P6,P7 P7,P8 P8, PI You probably noticed that the line direction is reversed in the description of the surfaces. The line vectors of surface I describe the surface as seen from the negative Z axis in a clockwise direction, while surface 11 is described in a counterclockwise direction. This small difference contains the solution to the hidden-line-problem. If you imagine the surfaces I and II as outer surfaces of a block, then SI is the front surface and SI I the rear surface of the block. The observation point is still on the negative Z axis. SI I is not visible from the observation point since it is hidden by the other surfaces. 58 Abacus Software ST 3D Graphics You can see that the description of the surface is always done in the clockwise direction from outside the cube and looking toward the current surface center. For the definition of the surface one wanders around the object to be described and determines the direction of the connection lines of the points belonging to the surface. As one can see in the next illustration, the visibility of the surfaces can be determined through the direction of the connection lines with a little vector algebra. To do this, start from any point on the surface and form the vector to the next point P= [px,py,pz] = [p2x-plx,p2y-ply,p2z-plz] , and the vector to the next point Q [qx, qy, qz] = [p3x-plx, p3y-ply, p3z-plz ], as well as the projection vector from a point on the surface to observation point A. An appropriate selection is the point PI,S[sx,sy,sz] = [ax-plx,ay-ply,az-plz]. As explained in the appendix, the product of two vectors (a\b) (see App. B) forms a vertical vector R= [rx, ry, rz ] =[py*qz-pz*qy,pz*qx-px*qz,px*qy- py*qx]. The direction of this vector results from the system in which the vector product was performed. In the left coordinate system used here, the vector d points in the same direction in which a screw with a left-handed thread would move from P to Q when turned, that is, it points with surface I in the direction of the positive Z axis and with surface II in the direction of the negative Z axis. Now we can say this about the visibility of surface I: if the vectors S and R are pointing in the same direction, the surface is visible from the observation point. If the vectors S and R point in different directions, the surface is not visible. As mentioned earlier, this process is limited to closed convex bodies, but the error is not very large with concave bodies. 59 ST 3D Graphics Abacus Software Figure 2.7.3-4: Hardcopy of bodies before and after Hidden-Line-Algorithm 60 Abacus Software ST 3D Graphics Figure 2.7.5 61 ST 3D Graphics Abacus Software Figure 2.7.6 Figure 2.7.7 62 Abacus Software ST 3D Graphics The error with concave bodies is that surfaces which are visible from the observation point are hidden by other surfaces but are not recognized. Now only the "direction comparison criterium" between two vectors is missing. This is accomplished by the scalar product of two vectors (S *R) which is defined as follows: c = |SI * IRI*cos(Phi) = sx*rx+sy*ry+sz*rz cos(alpha) > 0 Figure 2.7.8 c is a real number and phi is the angle enclosed by S and R. From Figure 2.7.8 we can see that the vectors a and b point in the same direction when cos (phi) is positive. The recognition of hidden surfaces can be summarized as follows. 1. Creation of a surface list in which the points are listed in a clockwise direction. 2. Finding the vectors P and Q from three successive points for each surface. 63 ST 3D Graphics Abacus Software 3. Determination of the vector S [sx, sy, sz ] from a point on the surface to the observation point. 4. Determination of the vector perpendicular to P and Q R [rx, ry, rz ] through the vector product (P\Q). 5. Comparison of the direction of the vectors S and R by checking the sign of the scalar product (S*R) through multiplication of the single components from S and R (Scalar product = sx*rx+sy*ry+sz*rz) 6. Marking of surfaces which have positive scalar products as visible surfaces. (Applies to left coordinate systems. In right coordinate systems the surfaces with negative scalar products are visible surfaces.) 7. Drawing the visible surfaces. 64 Abacus Software ST 3D Graphics 2.8 Rembrandt and hidden surfaces You probably want to know what computer graphics and a painter who died in 1669 have in common. An oil painting is created from back to the front, that is to say, the painter first draws the background and then objects are placed further to the front simply by covering the background with oil paint. This method, carried over to the computer, is another solution of the hidden surface problem. A middle Z coordinate is calculated for each surface and, as an example, all Z coordinates of the comer points can be added and divided by the number of comer points which are stored for the surface. Then the surfaces are sorted according to size and drawn from the largest to the smallest Z coordinates. To insure that the surfaces which are painted over have really been covered, we can’t just to draw the outer lines of the surface. It is necessary to fill the surfaces with color. The surface construction from the back to the front is shown in the following illustrations. Figures 2.8.1-5: Hardcopy of the surface construction Figure 2.8.1 65 ST 3D Graphics Abacus Software Figure 2.8.2 Figure 2.8.3 66 Abacus Software ST 3D Graphics ST 3D Graphics Abacus Software Figure 2.8.6 Figure 2.8.7 68 Abacus Software ST 3D Graphics Figure 2.8.8 Of course, the two methods for the removal of hidden surfaces can be combined. First the visible surfaces can be determined through scalar products. Followed by sorting the surfaces according to descending Z coordinates, and then drawing them. 69 ST 3D Graphics Abacus Software 2.8.1 Light and Shadow In general, there are two types of illumination, direct and indirect. With indirect illumination the intensity of the light is equal on all places in space. The indirect light is created through diffuse reflection from other objects, such as walls and ceilings. The appearance of an object in space under this illumination is dependent only on the reflection coefficient of the object. This reflection coefficient is the relationship of reflected light rays to the total striking the surface. Its value runs from zero for a black body (all light rays which strike are absorbed) and one for a white body (all light rays which strike are reflected). A body whose reflection coefficients are between zero and one is designated as a gray body. A reflection coefficient R can be given for every surface which determines the intensity of the surface. Intensity = R * IL with IL = Intensity of available indirect light. A more realistic representation results from the definition of one or more point light sources in the space. These point light sources, for example lamp, candle, or sunshine, have a certain position in the space and shine in the direction of the object. In this case, the orientation of the illuminated surface to the light source is of great importance. More light rays fall on a surface which is perpendicular to the light source than an equally large surface which is not perpendicular to the light source. The orientation of the surface to the light source can be determined by comparing the normal vector of the surface (the vector perpendicular to it) with the vector to surface from the light source. If L and N are two vectors of length 1, the relation for the angle between L and N is: L*N = lx*nx+ly*ny+lz*nz = cos(w) For the gray value of the surface the result is then: Intensity = R*IL + R* (L*N) *DL with the reflection coefficient R and the intensity of the direct light source DL, which is between zero and one. 70 Abacus Software ST 3D Graphics Figure 2.8.9: Surfaces with Light Rays 71 r Machine Language Fundamentals for Graphic Programming Abacus Software ST 3D Graphics 3. Machine Language Fundamentals for Graphic Programming All programs described in this book may be run on various ST computer/monitor combinations. To simplify the compatibility, all drawing functions for the 3-D graphics project were done with operating systems functions (line-A). To introduce you to machine language programming on the ST, we first have an explanation of some of the basic principles (sine) and then a small program for drawing random lines. This program illustrates the program interface to the operating system and a simple line-drawing algorithm which writes directly to the screen. The line-drawing algorithm is not necessary for the 3-D project coming later and is intended only as an example. The use of the algorithm is limited to monochrome monitors. Owners of color monitors can replace the call drawl with ddrawl (indicated in the listing) if they want to run the program mainl. s. 75 ST 3D Graphics Abacus Software 3.1 Speed Advantages from tables Before starting a project in machine language, you should think about the number format to be used. For all the following applications we can perform all calculations with 16-bit integers. Another problem is the sine function, whose function values can range from -1 and +1. The function values can be approximated on computers using the Taylor series, which approximates the exact function value through repeated summation of the terms of a sequence. In practice, the summation can be terminated after 3 or 4 terms. As an example, we have here the Taylor series for the sine function. sin(x) = x - xV3! + x 5 /5! - x 7 /7! + ... The angle x is given in radians, and 3! means 3 factorial = 1*2*3 = 6. This method is not suitable for quick calculation of sine and cosine values because several multiplications must be performed for each function value. A rather unelegant but simple and common solution is to store all the necessary function values in a table in memory, which can then be accessed very quickly. The accuracy can be set as desired since the function values are calculated before the actual program application and the time factor does not play a role. In our example, all sine values between 0 and 360 degrees are entered in steps of one degree. This is quite adequate for almost all applications which require trigonometric functions. Should an intermediate value be required, it can be interpolated from the table. Since the cosine function is the same as the sine function shifted by 90 degrees, the cosine functions can also be taken from the sine table. The function values of the angle functions are real numbers which are floating point numbers with several places after the decimal point. Since all our calculations involve only integers, it is necessary to transform the values of the sine function. This is done by multiplying by a sufficiently large number-in our example with 214 = 16384. 76 Abacus Software ST 3D Graphics The length of the line c and the angle alpha are already known, and we want to find the length of b. According to the definition of the angle function, the length of the distance =c* sin (a) = 20*sin(45). The sine of 45 degrees is 0.707106781 with nine-place precision. In our table we have the value 0.707106781 * 16384 = 11585 for 45 degrees. After multiplying by 20 we got the number 231700 as a result. We don’t have to worry that this number will exceed the value range of 16-bit integer arithmetic because the processor always produces a 32-bit product as the result of a 16-bit multiplication. This 32-bit result, the number 231700, can now be adapted to the original value range by dividing by 16384, and we get 14 as the result. You may ask yourself why 16384 was used for the multiplication: first of all the number is large enough to extend the range of the sine function. Numbers between -1 and 1 become numbers between -16384 and + 16384. Second, the multiplication can be performed with two very fast commands of the processor. Multiplications by a multiple of two can be replaced in all microprocessors with shift commands which don’t take much more time than an addition. 77 ST 3D Graphics Abacus Software At this point I would like to briefly discuss the possibilities of the table representation in the computer. The sine table is the simplest form of a table, a linear list. The individual table values are stored sequentially in memory. Our sine table for the first values looks like this: sintab: .dc.w 0,286,572,857,1143,1428,1713,1997,2280 .dc.w 2563,2845,3126,3406,3686,3964,4240,4516 .dc.w 4790,5063,5334,5604,5872,6138,6402,6664 Since the gradations of the angles are in 1 degree steps, the first table value gives the sine of 0 degrees, the second the sine of one degree, the third the sine of two degrees, etc. The 91st table value is the sine of 90. table value and the sine of 360 degrees is represented by the 361st value. Zero is chosen as the start to match the table numbers to the corresponding angle. This means that table value zero represents the sine of zero degrees. Value number 90 corresponds to 90 degrees and 180 to 180 degrees. The 68000 computer makes access to this table very easy through its excellent addressing capabilities. The initial address of the table is loaded into the address register. This is the address where the zero element is stored. With the number of the desired table value in a data register it is possible to access the location using the addressing mode "address register indirect with index." In this table format it is absolutely necessary to pay attention to the data length of individual entries. The address of the zero value is equal to the beginning address of the table plus zero, but the address of the first value is the beginning address of the table plus two, since each value occupies two bytes. This means that the index number in the data register must be multiplied by the number of bytes for one entry. In this case it is two bytes. This multiplication by two is very fast with one left shift of the bits in the index number. 78 Abacus Software ST 3D Graphics 3.2 Assembler routines for screen manipulation The screen of the Atari ST is organized using what is called bit-mapped graphics. This means that bits which are set in the screen storage correspond directly to dots on the monitor and therefore there is no difference between text and graphics. Since the screen memory is part of the main memory of the CPU, it can be manipulated quickly, i.e. without waiting cycles. For monochrome display the resolution is 640*400 points, which are represented by 400 times 640 bits in RAM. Address: $78000 _ $78001 _ $78002 _ ■■■$7804F X 0>*= X >- 639 ^ $78000 76543210 76543210 $78050 Bil number $780A0 $780F0 $780F1 $7FCB0 1 Y 0 >- Y >- 399 r Figure 3.2.1 The only routines required for screen manipulation are those for displaying a point and for drawing and erasing lines. A line of the video picture is formed from 80 bytes and the total picture is made up of 400 lines. The address of a picture point can be calculated as follows: address = screen start + Y*80 + INT(X/8) 79 ST 3D Graphics Abacus Software The bit number of the byte can be obtained with the following formula: number = 7 - (X MOD 8) The function I NT truncates the positions after the decimal point of a real number, while the function MOD returns the remainder of the operand by the second. For example, 9 MOD 2 returns 1 as the result Screen start is the starting address of the screen memory, which is $78000 on the 520 ST and 8F8000 on the 1040 ST It may appear to be somewhat unusual to have the coordinate origin in the upper left comer, but it is easy to change to the lower left comer and this is accomplished by negating the Y values and adding 399. The X coordinates remain unchanged of course , since the zero point is already in the left comer of the display. The Y coordinate 370 in a normal left system becomes (-370+399) = 29 in the screen system. This conversion need be made only immediately before points are drawn. Some calculations are required to draw a single point. The speed advantage of tables for the calculation of the address of a point should also be considered here. This table holds the RAM address for every possible Y coordinate. This saves a multiplication for every calculation of the screen address. Since the plot-point routine is used very often for drawing lines, the speed advantage gained by using this table is correspondingly great. 3.2.1 Drawing lines Since the size of a point on the screen is dependent on the resolution of the computer, it is not possible to represent a line in the mathematical sense. A line which connects two points PI and P3, actually takes a more or less jagged path. 80 Abacus Software ST 3D Graphics T Xi Figure 3.2.2 Starting from point PI, you have the problem of deciding which points must be set, in order to reach point P3. Note that it is possible to set the points only at the intersections of the raster lines. The line is formed when either the X coordinate is retained and a point drawn with an incremented Y coordinate or you can increment the X coordinate while the Y coordinate retains its value. In mathematics, a line which connects two points is described through its slope m. m is a measure of the "steepness" of the line and the larger m becomes, the steeper the line becomes. With a positive m, the line rises from left to right, while with a negative m it slopes down from left to right For a line parallel to the Y axis, the slope is infinite. The expression for the slope: m = dy / dx 81 ST 3D Graphics Abacus Software Figure 3.2.3 See Figure 3.2.4 for an explanation of the algorithm for drawing of lines. 82 Abacus Software ST 3D Graphics Figure 3.2.4 Let us assume that in drawing the line from PI to P3 that we have already reached the point P2 already and now have to decide the direction in which to draw. In our example, the point P2 is ’’over" the ideal line from PI to P3. Expressed mathematically, the slope of the connecting line from Point PI to P2 ml=(p2y-ply)/(p2x-plx) = wy/wxis greater than the rise of the line which connects the points PI and P3 m2= (p3y-ply) / (p3x-plx) =dy/dx. As the illustration shows, the next step in drawing must be made in the X direction. With the comparison of the two slopes, we have found a decision criterion for the direction of drawing: If the slope of the connecting line between the starting point of the drawing PI and an intermediate point P 2 is greater than the slope of the line between the beginning and end points (PI, P3), a drawing step should be made in the X direction . If the slope is smaller, the next point should be drawn in the Y direction. For the purpose of programming this criterion we shall define a decision variable D, which is assigned the difference between the desired and the actual slope. 83 ST 3D Graphics Abacus Software D = (dy/dx)-(wy/wx) If D is larger than zero ==> Step in Y direction If D is smaller than zero ==> Step in X direction After a small conversion we get: D*dx*wx = (wx*dy)-(wy*dx) Multiplications slow down calculations, so we should try to eliminate them from the calculation. The exact value of D is of no interest. It is only important to know how D changes with a step in the X or Y direction so that an eventual change in the sign of D can be recognized. For this reason it is also possible to replace the expression D*dx*dy with D again. D = (wx*dy) - (wy*dx) During a step in the X direction, wx is increased by one while we retain the old value of wx. For our D which we call new D or ND to distinguish it from D, the following results: ND = (wx+l)*dy - wy*dx ND = wx*dy + dy - wy*dx The last expression is equal to old D + dy, where old D corresponds to the value of D before the step in the X direction. Analogous to this for a step in the Y direction: ND = wx*dy - (wy+l)*dx ND = wx*dy - wy*dx - dx As you can see, D is reduced by dx with a step in the Y direction. For ND can be written: Step in Y direction ND = D - dx Step in X direction ND = D + dy 84 Abacus Software ST 3D Graphics The multiplications have been replaced according to our desires by additions. To formulate the algorithm, we must still decide in what direction we will draw if D is zero. This can be decided at random and in our example ND=0 results in a step in the Y direction. Another special case which has not been mentioned is when dy is zero. In this case, steps can be made only in the X direction since the resulting line must be a parallel to the X axis. This case can only be determined with a test at the beginning of the routine. Furthermore, we have only considered lines with a positive slope, that is, those where py3 is smaller than pyl. To retain the decision method in this form, it is necessary to make negative dx and dy values positive through multiplication with -1, and to decrease the X and Y coordinates by one instead of increasing them for every step in the X or Y direction. The algorithm for drawing a line between the points PI [xl, yl] and P 3[x3,y3] appears like this in a structogram: 85 ST 3D Graphics Abacus Software Figure 3.2.5: Structogram Draw line 86 Abacus Software ST 3D Graphics 3.3 Operating system functions Since we will use only operating system functions for the 3-D graphics programming, some should be explained before they are used. One of these functions is the routine for switching the beginning address for the video controller. All computers which which can display animated graphics quickly and flicker free have the ability to work with two logical screen pages internally. Fast drawing and erasing of objects on the screen and the rapid accesses to the screen RAM by the computer and the video controller, causes the monitor picture to be unstable and to flicker. If the hardware has the ability to tell the video controller where in RAM the screen memory starts, the strategy for the creation of flicker free graphic is very simple. We define two logical screen pages. We will use the Atatri ST as a concrete example: in the Atari ST with 512K RAM the standard screen page is stored between $78000 to $7FFFF and it is possible to define a second screen page from $70000 to $77FFF. In the initial state, both screen pages are erased and the video controller shows the page starting at $78000. Now the first picture can be drawn in the RAM starting at address $70000. After drawing the picture, the video controller is informed at a suitable time of the new beginning address for the screen RAM ($70000). A suitable time for switching is the time period in which the electronic beam which draws the video picture returns, without being seen, from the lower right comer of the screen to the upper left corner. This moment is even recognized by the operating system and the switching of the screen pages can be solved without any major programming effort. If the page starting at $70000 is being displayed by the video controller, the CPU can draw another picture, such as the object in another position, in the page starting at $78000 without disturbing the picture construction. After the new picture is completed, pages are switched again and you can erase the old picture in the storage area which is not being displayed. In general, the page which is being displayed is considered to be the physical page in which the drawing is taking place is the logical page. Only when both are identical do you see the progress of the drawing on the screen. 87 ST 3D Graphics Abacus Software 3.3.1 Starting a Program To start a machine language program on the Atari ST you have to know what happens when a program icon is clicked with the mouse. The operating system loads the appropriate program and passes control to the program once it is loaded. After loading a program, the operating system declares the entire memory as occupied so that it is not possible to move data or program sections. To avoid this disadvantage, the called program must determine its actual memory requirements, declare this area as occupied, and leave the rest of the memory free. The Atari operating system simplifies this task by passing a pointer on the stack to the called program indicating the memory area occupied by the program and data. The called program can calculate the memory actually required and declare the unused area as vacant to the operating system. Note: sufficient space must be reserved for the processor stack. From the Digital Research documentation, it is not clear how much stack space is required for the GEM functions, but the 4K bytes reserved for this purpose in the example should be sufficient for all purposes. To make it possible to use all GEM functions, it is recommended that the program call the functions Application-Init and then Open-Virtual- Workstation when it starts. After these two calls, GEM-DOS, the BIOS, Extended-BIOS and the AES and VDI functions are available to the program. An overview of these functions are available in the two Abacus Software books Atari ST Internals and the large Atari ST GEM Programmer’s Reference. All programs in this book were written using the assembler from Digital Research. For users of other home computers the assembler is probably new, and so I want to discuss it briefly. The assembler is completely disk oriented, i.e. all input and output comes from and goes to the diskette. First you create the source text of the program with an editor, store it on a diskette and call the assembler with name of the source text The assembler processes the source text by creating several auxiliary files on the diskette. Finally it writes the desired object file on the diskette. The object file which was created, recognizable by the extension . o, is not executable since it was assembled at the absolute address zero. To generate an executable program the absolute addresses must be replaced with relative addresses to make it possible to load the program into any memory area. For this purpose, you call the program RELMOD.PRG 88 Abacus Software ST 3D Graphics which then creates the desired run-time program file. In this you can write manner machine language programs whose length is limited only by the storage capacity of the computer and the floppy disk. It is impossible to combine two programs which are already object files with this method, however. For this reason, one usually adds an intermediate step, as is also done with higher level languages, called linking. The linker permits several separately-assembled object files to be combined into one single file. Large assembly language programs quickly become difficult to understand and it is recommended that they be divided into at least two modules. The first module initializes the program and contains all of the error-free and tested subroutines, while the second module contains the latest main program. This can reduce the assembly time considerably since the large basic module must be assembled only once and afterwards only linked to the main program. The use of the linker also permits the use of assembler directives which would otherwise not be possible. The assembler in conjunction with the linker can manage three separate program areas: text, data, bss. The text area contains the actual program, i.e. the program text, and the data area contains the initialized data. These are variables to which values were assigned already before the start of the program. In the bss area there is storage space reserved for the data which has not been initialized. Each of the programs; assembler, linker and relocator require parameters, which are passed during the start. To assemble the basic module, first select AS68.PRG and then INSTALL APPLICATION from the OPTIONS menu as TOS-takes parameters. Then enter the following line into the dialog box which appears: -p -I -u basicl.s > basicl.lst where basic 1. s is the name of the text file to be assembled. The -p and > basicl.lst statements create a listing to the disk of the assembly process which can later be printed for examination. The assembler creates a file with the name basicl.o. This object file contains the tested subroutines and will be linked to the current main program. 89 ST 3D Graphics Abacus Software To assemble and link the main program, it is best to create a batch file, which contains the individual command sequences. The batch file could look like this: as68 -1 -u %2. s wait.prg link68 [u] %2.68k=%l.o,%2.o relmod %2.68k %2.prg rm %2.68k rm % 2.o wait.prg This batch file might be stored under the name aslink.bat on the diskette. The batch file is made very flexible through the use of two place holders, %1 and %2. To assemble the main program with the name mainl. s and the subsequent linking with the basic module basicl. o You call the program batch.ttp and pass the command sequence in the dialog box: aslink basicl mainl After the assembly process the desired program file mainl.prg is finally on the diskette. This creation of modules makes working with the disk drive more bearable and the coffee breaks during assembly shorter. As a practical test of all this, we have here the first version of the basicl . s program and the first demo program. The basic program contains only the initialization of the program and the basic routines for screen manipulation such as screen erasing, and drawing of points and lines. Assembly is done with: as68 -1 -u basicl.s The first main program demonstrates the speed of the computer by drawing random lines and demonstrates how to call the operating system. The steps for the creation of the ready-to-run program file mainl .prg, without using a batch file are as follows: 1. Assemble MAIN1.S with the AS68.PRG. 90 Abacus Software ST 3D Graphics 2. Link the two object files with the Linker. Iink68 [u] mainl.68k = basicl.o,mainl.o 3. Create a relocatable program with relmod mainl.68k mainl.prg The file mainl. prg can be started by clicking with the mouse after the file Relmod, the two files mainl. o and mainl. 68k, which are no longer needed, are erased with the program RM. The listing should be self-explanatory with all of its comments. It should offer an easy introduction to graphics programming in machine language. More detailed explanations of the routines used can be found with the explanation of the link files grlinkl.s in section 4.1. Starting with Chapter 4 we will really start to program. 91 ST 3D Graphics Abacus Software *********************************************************************** * Link file basicl.s, is linked with the main program whose entry * * routine must have the name main. * * U.B. 11.85 * *********************************************************************** .globl wait,wait1,drawl,ddrawl,inlinea .globl grafhand •globl grhandle .globl global,contrl,intin,intout,ptsin,ptsout,addrin,addrout .globl apinit,openwork,clwork,aes,vdi .globl inkey .globl mouse_on,mouse_off,printf .text ******************* * Entry to the program, initialization of all operating system * * functions and creation of the Y-tables (For computers with color * * monitors, replace "jsr startl" with "jsr start2". * * Furthermore when using a color monitor/ replace all * * "jsr drawl" calls in the main program with "jsr ddrawl". ********************** art: ★ initialize the program movei1 a7, a5 ★ Base page address is on the stack move.1 A (a5),a5 * base page address = program start - $100 move.1 $c(a5),dO * Program length add. 1 $14(a5),dO ★ Length of initialized data area add. 1 $lc(a5),dO ★ Length of data area not initialized add. 1 #$1100,dO * 4 K-Byte user stack=sufficient space move.1 a5,dl ★ Starting address of the program add. 1 dO, dl * Plus number of reserved bytes = space required and. 1 #-2,dl * even address for stack move.1 dl, a7 * User stackpointer to last 4K- byte move.1 dO, -(sp) * Length of reserved area move.1 a5, -(sp) * Beginning address of reserved area move.w dO,-(sp) ★ Dummy-Word move.w #$4a,-(sp) ★ GEM DOS function SETBLOCK 92 Abacus Software ST 3D Graphics trap #1 add. 1 #12,sp ★ old stack address restored again jsr start1 * Create Y-table jsr main * Jump to main program. ( User-created move.1 #0,- (a7) * Terminate program running trap #1 ★ Back to Gem-Desktop ******************************************************************** * Call a AES-Routine, where the parameters are passed to * * to the various arrays (contrl,etc.) * ******************************************************************** aes: move.1 #aespb,dl * call the AES routines move.w #$c8,d0 trap #2 rts ************************************************************************ * Call a VDI-Routine * ft******************************************************************** vdi: move.l #vdipb,dl * call the VDI routines move.w #$73,dO trap #2 rts ********************************************************************** * Announce the program * ********************************************************************** clr. 1 dO move.1 dO,aplresv move.1 dO,ap2resv move.1 dO,ap3resv move.1 dO,ap4resv move.w #10,opcode move.w #0,sintin move.w #1,sintout move. w #0,saddrout move.w #0,saddrin jsr aes rts * Announce the program as * Application 93 ST 3D Graphics Abacus Software *********************************************************************** * Check on screen handler and store for other functions * *********************************************************************** grafhand: move.w #77,contrl * Get the screen handler move.w #0, contrl+2 * and store it in the global move.w #5,contrl+4 * Variable grhandle move.w #0,contrl+6 move.w #0,contrl+8 jsr aes move.w intout,grhandle rts *********************************************************************** * Open a Virtual Screen Work Station where all GEM drawing functions * * will occur. * *********************************************************************** openwork: move.w move.w move.w move.w move.w move.w move.w move.w move.w move. w move.w move. w move.w move.w move.w move.w jsr rts #100,opcode #1, d0 #0,contrl+2 #11,contrl+6 grhandle,contrl+12 dO,intin dO,intin+2 dO,intin+4 dO,intin+6 dO,intin+8 dO,intin+10 dO,intin+12 dO,intin+14 dO,intin+16 dO,intin+18 #2,intin+20 vdi * open a workstation * screen handler 94 Abacus Software ST 3D Graphics *'*******★★*******★***★**★★★**’***★***★*★★**★**★***★**★★**★★★★★*******★★* * Clear the workstation * *********************************************************************** clwork: move.w #3,contrl * Clear workstation move.w #0,contrl+2 * clear the screen move.w #1,contrl+6 move.w grhandle,contrl+12 jsr vdi rt s * Turn on the mouse and its control. * move. w #122,contrl * turn on the mouse move.w #0,contrl+2 * its control move.w #1,contrl+6 move.w grhandle,contrl+12 move.w #0,intin jsr rts vdi *********************************************************************** * Turn off the mouse and control. * *********************************************************************** mouse off: move.w #123,contrl * turn off the mouse move.w #0,contrl+2 * its control move.w #0,contrl+6 move.w grhandle,contrl+12 jsr vdi rts 95 ST 3D Graphics Abacus Software ********************************* * Write a string on the screen ********************************* printf: move. 1 aO,- (a7) move. w #9,-(a7) trap #1 addq. 1 rts #6,a7 wait 1 dbra rts dO,waitl wait: move.w #1, -(a7) trap #1 addq.1 rts #2,a7 ************************************* * ************************************* * write the string, whose * beginning address is in * register AO, on the screen. * String must terminate with * zero. * Time loop, counts the dO-Register * down to -1 * wait for a key stroke * GEM-DOS-Call * Sense keyboard status (does not wait for keypress) and return key * * code and also the scan code. inkey: move.w #2,-(a7) ★ Sense keyboard, does not wait for key move.w #1,-(a7) * activation and return an ASCII-code trap #13 ★ of an activated key in the lower half addq.1 #4,a7 ★ of the long word of DO, and the scan code tst. w dO * in the upper half of the long word of bpl endkey ★ DO. move.w #7,-(a7) trap #1 addq.1 #2,a7 endkey: rts 96 Abacus Software ST 3D Graphics e******************************************************************** * Draw-line-routine, draws directly into the screen storage and is * * used only for the high resolution mode {640*400 Points ). For color * * monitor use ddrawl ************************************ drawl: move. 1 d7,-(a7) * move.1 #ytab,aO * clr. 1 d4 move. w #1, a4 * move. w a4, a5 * move. w a2,d6 sub. w d2,d6 * bge dxispos neg. w d6 * move. w #-l,a4 * dxispos: move .w a3,d7 sub.w d3,d7 * bgt plotit * beq dyis 0 * neg. w d7 * move .w #-l,a5 * bra plotit dyis_0: not. w d4 * plotit: tst .W d2 * bmi draw_it * tst. w d3 bmi draw_it cmp. w #639,d2 bhi draw_it cmp. w #399,d3 bhi draw_it move.w d3,d0 * lsl. w #2, dO * move.1 0(aO,dO.w),al * move.w d2,dl * lsr .w #3, dl * move.w d2,d0 * not. w dO * * ********************************** Save register Address of the Y-table X step = +1 Y step = +1 DX in d6 = X2 - XI If DX is negative, then make positive through negation DY in d7 If DY is larger than zero draw then first point DY is negative, make positive Y-Step is then -1 If DY = 0 then parallel to X-Axis Test if drawing area was exceeded Y-value times two for access to Plot table Screen address X-value INT (X/8) X-value -X 97 ST 3D Graphics Abacus Software ********************************** * Here the point is drawn * ********************************** bset dO, 0(al,dl.w) * 7-(X MOD 8) with the bset-command draw it: cmp. w d2,a2 * End X reached? bne notend * no cmp. w d3, a3 * End Y reached? beq endit * no notend: tst .w d4 ★ D > 0 => Y step bge ystep xstep: add. w a4,d2 ★ else X step X=X+-1 add. w d7,d4 ★ ND = D + DY bra plotit ystep: add. w a5, d3 * Y=Y +- 1 sub. w d6, d4 * ND = D - DX bra plotit drawend: endit: movem. 1 (a7)+,d7 * restore register rts ★ Return to calling program ********************************************************************** * This Draw-line-routine is universal for all monitor types and * * can be used with all resolutions. * ********************************************************************** move.1 d7,-(a7) move.1 tlineavar,aO move.w d2, 38 (aO) * XI move.w d3,40(aO) * Yl move.w a2,42 (aO) * X2 move.w a3,44(aO) * Y2 -dc. w $a003 * draw line move.1 (a7)+,d7 rts 98 Abacus Software ST 3D Graphics ********************************************************************** * Initialize the Line-A variables and store the address of the * * Variable block in lineavar. * ********************************************************************** inlinea: .dc. w $a000 ★ initialize the Line move.1 aO,lineavar move.w #0,32 (aO) move.w #$ffff,34 (aO) * Sample of the line move.w #0,36(aO) * Writing mode move.w #1,24(aO) * drawing color rts * Creation of the Y table for the highest graphic mode (640*400) * startl: move. w #2,-(a7) * checks the screen address of the trap #14 * System, recognizes which computer addq. 1 move.1 #2,a7 d0,physbase * Display start minus 32 K-Byte move. 1 #399,dl * Number of lines minus one move. 1 #ytab, aO * Physical address stloopl: move.1 dO,(aO)+ * New address equals old address add. 1 #80,dO * plus 80 dbra rts dl,stloopl * Line-A initialization * start2: jsr inlinea * Initialize line A rts 99 ST 3D Graphics Abacus Software ******************************************************************** * Variables of the basic program * ******************************************************************** lineavar: .even .bss .ds. 1 1 * Storage for address of Line-A variable physbase: .ds. 1 1 * Storage for screen address. ytab: .ds. 1 400 * Storage for the Y table contrl: * Arrays for AES and VDI functions opcode: .ds .w 1 sintin: .ds.w 1 sintout: .ds. w 1 saddrin: .ds.w 1 saddrout: .ds.w 1 .ds.w 6 global: apversion: .ds.w 1 apcount: .ds.w 1 apid: -ds.w 1 apprivate: .ds. 1 1 apptree: .ds. 1 1 aplresv: .ds.l 1 ap2resv: .ds. 1 1 ap3resv: .ds.l 1 ap4resv: .ds.l 1 intin: .ds.w 128 ptsin: .ds.w 256 intout: • ds.w 128 ptsout: .ds.w 128 addrin: .ds.w 128 addrout: .ds.w 128 grhandle: .ds.w 1 vdipb: .data .dc. 1 contrl,intin,ptsin,intout,ptsout aespb: .dc. 1 contrl,global,intin,intout,addrin,addrout .end 100 0 Abacus Software ST 3D Graphics *********************************************************************** * Main program for link file basicl.o , runs only in connection with * * this link file . U.B. 11.85 * Draws random line in coordinate area 0-255. The value area * is valid for both axis *********************************************************************** .globl main .text * Entry point from the linkfile * main: jsr apinit jsr grafhand jsr openwork jsr mouse_off jsr clwork W jsr inlinea loopl: jsr clwork move. 1 ttextl,aO jsr printf move.1 loopc,d7 loop2: jsr random and. w border,dO move.w dO, xO jsr random and. w border,dO move. w dO, yO jsr random and. w border,dO move.w dO, xl jsr random and.w border,dO move.w d0,yl move.w x0,d2 move.w xl,a2 move.w y0,d3 * Announce application * * Open screen work station * Hide the Mouse * Clear Display * Color version only * Address of text after AO * Write text * Generate random number * bring to area 0-255 * through masking out of the upper * 8 Bits of the lower word in DO * transfer the two points to the * "right" registers 101 ST 3D Graphics Abacus Software move.w yl,a3 jsr drawl * Draw line from XO,YO to XI,Y1 sketch dbra d7,loop2 * Repeat loopc jsr inkey * Sense keyboard, do not wait for key swap dO * activation, scancode in DO cmp. w #$44,dO * compare with code in F10 bne loopl * If not : loop again rts * otherwise terminate program *********************************************************************** * Call the operating system function for creation of a 4-byte integer* * random number, the number is returned to DO. * *********************************************************************** random: move.w #17,-(a7) * generate a 4-Byte Integer- trap #14 * Random Number in DO. Use only addq.l #2,a7 * the lower 2-Bytes rts .data .even ********************************************************************** * Variables for the Main program * * * ********************************************************************** ********************************************************************** * Text for the printf function, 27 Y 34 96 positions the cursor * * Sequence is column, line, both with an offset of 32 * ********************************************************************** text 1: . dc -b 27, ' Y' ,40,42, ' Random lines ',0 loopc: . dc. 1 60 * Number of lines border: . dc. w $ff * 255 as display limit, with the high' ★ * resolution B-W monitor the $ff * * can be replaced with $lff = 511 102 Abacus Software ST 3D Graphics ********************************************************************** .bss .even x0: .ds .w 1 * Temporary storage for the two Y 0: .ds.w 1 * Points, the program runs with small xl: • ds .w 1 * changes even without the intermediate yl: .ds.w 1 * storage; what changes ? .end 103 Abacus Software ST 3D Graphics 4. Graphics using assembly-language routines The programs presented in the following part of the book can be used with monochrome as well as color monitors, since the line drawing is performed by the operating system, or to be more accurate, by the LINE- A-routines. Of course it would be possible to convert the draw-line- algorithm from the first link file for the various picture formats, but this process has the disadvantage of requiring a subroutine for every picture format. The programs described here can be executed on all kinds of computer-monitor combinations. During program start, the main program recognizes what type of monitor is attached and what resolution is desired and on the basis of this information provides some variables with the required data. For example, the coordinate origin of the picture system is placed in the middle of the display. The larger memory capacity of the ST permits it to handle significantly larger quantities of data. Once the operating system of the smaller models is placed in ROM, the area released in RAM will be sufficient even for the largest applications. When calling the Metacombco Editor for input of the larger source files (grlinkl, menul, rotatel, paintl) you have to specify more memory space for listing to be entered along with the filename. To do this, enter grlinkl. s 23000 in the dialog box that appears. This reserves about 80k for the source text. If you enter source text without comments the space reserved in the basic version of the editor should be sufficient. 107 ST 3D Graphics Abacus Software 4.1 Definition of a data structure for an object in space The program modules presented here have the ability to represent on the screen any object in a user defined world in any position, as seen from various positions. The single disadvantage is the limitation of the valid value range to ±32000; this means that for the definition of the world a right angle three dimensional Cartesian coordinate system (right system) is available whose three coordinate axes (X-Y-Z) are labeled with values between +32000 and -32000. Whether these values are in meters, kilometers or the number of corrupt politicians in the Senate depends on the individual user and the application. For example, using the number of corrupt politicians is a questionable value, since it changes from moment to moment, and is far from constant. Joking aside, a very simple object should suffice to describe the data structure. We will use simple house as in Figure 4.1.1. Figure 4.1.1: House as Wire Model 108 Abacus Software ST 3D Graphics Every object in the coordinate system is described through a finite number of points and the lines which connect these points. To represent the object, these points in the world system must be specified by declaring of their coordinates. It has proved to be useful to define the object, in this case the house, in its own coordinate system and to transform it during the construction of the world coordinate system. To gain an advantage, the coordinate origin of the object system is located inside the house, if possible at a "rotationally neutral” point, i.e. during a rotation of the object around this point, the maximum changes of the individual points resulting from the rotation should be minimized. The object should not be distorted. Figure 4.1.2: House with coordinate system included The individual steps during the "construction” of the house therefore are: 1. Draw a total view of the object (on a piece of paper) and arbitrarily number of the individual points. 109 ST 3D Graphics Abacus Software Figure 4.1.3: House with numbered points 2. Draw the object in the various possible aspects with the current coordinate axis for accurate specification of the points. 110 Abacus Software ST 3D Graphics Figure 4.1.4 - Figure 4.1.9: six views of the house ill ST 3D Graphics Abacus Software Figure 4.1.6 Figure 4.1.7 112 Abacus Software ST 3D Graphics Figure 4.1.9 3. Set up a coordinate list of the individual points. 4. Create a line list, i.e. state which points are connected by lines. 5. Indicate the number of points and lines in the object. 113 ST 3D Graphics Abacus Software Coordinate list of the house: Point No. X-coord. Y-coord. Z-coord. 1. -30 30 60 2. 30 30 60 3. 30 -30 60 4. -30 -30 60 5. 30 30 -60 6. -30 30 -60 7. -30 -30 -60 8. 30 -30 -60 9. 0 70 60 10. 0 70 -60 11. -10 -30 60 12. -10 0 60 13. 10 0 60 14. 10 -30 60 15. 30 20 40 16. 30 20 10 17. 30 0 10 18. 30 0 40 19. 30 20 -10 20. 30 20 -40 21. 30 0 -40 22. 30 0 -10 23. 30 -10 0 24. 30 -10 -20 25. 30 -30 -20 26. 30 -30 0 Total of 26 points. 114 Abacus Software ST 3D Graphics Line list: Line No. from point to point 1. 1 2 2. 2 3 3. 3 4 4. 4 1 5. 2 5 6. 5 8 7. 8 3 8. 8 7 9. 7 6 10. 6 5 11. 6 1 12. 7 4 13. 9 10 14. 1 9 15. 9 2 16. 5 10 17. 6 10 18. 11 12 19. 12 13 20. 13 14 21. 15 16 22. 16 17 23. 17 18 24. 18 15 25. 19 20 26. 20 21 27. 21 22 28. 22 19 29. 23 24 30. 24 25 31. 25 26 32. 26 23 Total of 32 lines. 115 ST 3D Graphics Abacus Software Additional information on the object is required for the transformation of the house into the world coorinate system: the angles housxw, housyw, houszw, which describe a rotation of the house about one of the three axes in regard to the coordinate origin, and the location of the house in the world coordinate system. The location is the point to which the coordinate origin (rotationally neutral point) of the house system is displaced in the world system, housxO, housyO, houszO. In our first example program the coordinate origin of the house system is moved to the coordinate origin of the world system, housxO etc. and therefore zero. For further information, we need an observation point and a projection center, where both points naturally are described in world coordinates. In the simplest case the observation point is the coordinate origin point of the world system, and the projection center [prox, proy, proz ] is located on the positive Z axis of the world system. The system of the observer (view system) is a right system in our programs and it is not necessary to transform to a left system, to multiply all Z values by -1. For our case this means that after transformation the view system a point with the coordinates [ 10 , 10 ,- 300 ] is farther from the observer than a point with the coordinates [10,10,-200]. Furthermore, we need the normal vector (direction vector) of the projection plane. For simplification we assume that it is pointed from the projection center toward the coordinate origin of the world system and points toward the negative Z axis. The projection center lies on the Z axis and therefore has the coordinates [0,0, proz ], since the normal vector of the projection plane points in the direction of the negative Z axis, the rotation of the observation direction vector to the negative Z axis is not necessary. To help explain the coordinate systems and viewing points, we have here Figure 4.1.10 with the world system and the observation factors defined in it. 116 Abacus Software ST 3D Graphics Figure 4.1.10 Summary: To represent the house on the screen we need a total of four coordinate systems, where the various coordinate systems exist only in theory and all transformations occur in a single system. The defined points are stored in arrays in which the various coordinate systems are then reflected so they do not disappear after a transformation. The following coordinate systems are used: 1. The data system (housdatx, housdaty, housdatz), in which the house is defined at construction. 2. The world system (wrldx, wrldy, wrldz), in which, for example, a village is represented by several houses, where the houses are all created by transformation at various places in the world system from the one single house defined in the data system. 117 ST 3D Graphics Abacus Software 3. The view system (viewx, viewy, viewz), which is used for the description of the view transformation. The view transformation is the transformation into the observer system, which is described through the observation reference point and projection center as well as the vectors from projection center to the observation reference. The vector from the projection center to the observation reference point is therefore the normal vector of the projection plane. 4. The screen system has only two dimensions. The only transformation which occurs in this system is shifting the coordinate origin to any desired location with reversal of the Y axis. Something we also used in our example is the displacement to the screen center but other locations are also possible depending on the application. After this simplified observation situation, now an example for the general view-transformation of a more complicated model. As a fictional example we will use a world system which represents an airplane standing on a runway. The observation point of the system should be in the middle of the cockpit window, which is therefore the projection plane, and the eye of the pilot should be the projection center. Let us also imagine a tanker truck and an airplane hangar at some distance from the airplane. Two types of transformations are possible. 1. A transformation of the object, which might mean that the tanker truck moves and approaches the airplane, for example. In this case the movement must occur in the world system and only the coordinates of the tanker truck need be recalculated in the world system. 118 Abacus Software ST 3D Graphics 2. A movement of the observer, in this example the aiiplane. Let’s go back to the starting position and assume that the tanker truck remains in its original position. Now the airplane should move, and for simulation of this movement all objects in the world system, the tanker truck and the hangar, must be transformed. The entire world system would be rotated about the center of the cockpit window. For a left rotation of the airplane, everything must be rotated to the right. This connection can be easily verified: If you move your head to the left, the observed objects move to the right out of our field of view. When the airplane is moved without rotation the observer gets the impression of movement through the displacement of the coordinate origin of the world system before the projection. 3. A movement of the observer and the object, which means first a transformation of the truck in the world system and subsequent transformation of the total world system into the view system. Only after completion of these various cases do we get to the perspective transformation, i.e the projection from space to the projection plane, or more precisely into the screen. This was an example of a more complex observation model. You will probably ask why things have to be complicated by using an additional coordinate system, the view system, when we could do everything in the world system. This is true, but the addition of the view system improves the accuracy and provides for a better overview of the total system. Because of the rounding errors from the many transformations, our world of the tanker truck and the hangar, would according to the law of increasing entropy, degenerate to a chaotic mess after a few hundred transformations. The aspect of the better overview is at least as significant as the accuracy and I want to try to demonstrate this fact again. As you will see, all transformations can be carried out with a single routine. Our application combines almost all of the rotations with matrix multiplication and performs displacements before and after these multiplications. The displacements are not included in the matrix multiplications and our point coordinates are therefore not extended 119 ST 3D Graphics Abacus Software coordinates but consist only of the three coordinates [x,y,z] of the current point. The only routine used is the rotation around any selected point. As a reminder, during the rotation around any point, the coordinate origin must first be moved to this point, then rotated by the desired angle and finally the coordinate origin moved inverse to its original place by back transformation. For the sequence of our routine this means that the point about which rotation should occur is passed, also the rotation angles around the corresponding axes (xw,yw,zw). The rotation routine first calculates a multiplication matrix through multiplication of the rotation matrixes belonging to the various angles. Then all points belonging to the desired object [ x, y, z ] are manipulated in the following sequence: 1. The point [ x, y, z ] is moved to the rotation point. This is achieved through subtraction of the coordinates of the rotation point from the object coordinates. The result of this operation is the point [ x', y', z' ]. 2. The point [x', y' , z' ] is multiplied by the previously calculated total rotation matrix. Result: [x'' , y' ', z'' ]. 3. The point is transformed back to the "old" coordinate system by adding of the coordinates of the rotation point In this model the axes are not scaled. The size manipulation of the objects, i.e. their pictured size on the screen, is performed during the projection through movement of the projection plane. If different values are selected for the subtraction occurring at the beginning and the concluding addition, the movement of an observer in the world system is simulated. If the angles of the normal surface vector in relation to the world system are provided (in section 2.5 we calculated the angles through projection on the various surfaces) the position of the observer can be determined in space through one point and three angles. Let us assume that a person is moving our world system, where the house discussed in the first example is located at the coordinate origin. The eye of this person, or actually the retina of the eye, is the projection plane. It is irrelevant that the projection center in the human eye is in front of the projection plane, since the reversed picture resulting from this is turned around by the brain. For the simulation of this moving observer the 120 Abacus Software ST 3D Graphics coordinate origin of the world system must be moved to the center of the retina, but we are limiting ourselves to a single eye. The coordinates of the eye in the world system must be known; furthermore the head of the observer can be moved through three different axis. You can easily determine the axis yourself. The rotation about the first axis in our coordinate system corresponds to the X axis, described by the observer nodding his head up and down. The Y axis rotation is shaking his head. The head rotates on the Z axis when the observer attempts to touch his ear to his shoulder. If the three rotaion angles are known, the coordinate origin will be rotated about this angle and the observed object lies in the coordinate system of the observer. It is not necessary to reverse the movement of the coordinate origin which is similar to the example of the airplane. In principle, an unlimited number of displacements, rotations, and observer situations are possible: rotation of the house, rotation of the total system around one point, or any axis, and also the displacement with rotation into the observer system. To bring some order into this flood of rotations, our programming examples are limited to one fixed observer location point. This is no limitation on the observed effects on the screen however, i.e. in principle it is the same whether one assumes that an object rotated around a point, or the observer moved his head, provided the size relationships are suitably adjusted. Finally, the programmer must know the desired effect. There are many ways to achieve the same effects. And now, the description of the transformations of our data structure for the first, fairly simple transformation program. The concrete object (house) is defined in a coordinate system (housdatx, housdaty, housdatz). During the initialization of the program, the subroutine makewrld moves the house to any desired location in the world system (wrldx, wrldy, wrldz), with possible rotation. In the first program it is moved to the point [0,0,0] without rotation. All further operations concerning the house relate only to the world system. For example, the house can be rotated around any point of this world system, or only the position of the house can be changed by displacement. But now the initial scenario of our model changes through these transformations, so we store the data for the rotation of the world system in a new coordinate system (viewx, viewy, viewz), where the initial scene (in wrldx, wrldy, wrldz) is available at any time and can be reproduced at any time. 121 ST 3D Graphics Abacus Software After each operation in this world system, the coordinates of the displaced house are stored in the view system. The object should also be displayed on the screen. To do this, it must be adapted to the perspective of the viewer situation. In our example, the projection center is at the coordinates [0, 0, 1500]“therefore on the positive z axis of the right handed coordinate system. Through the perspective transformation, the coordinates of the view storage are transformed into screen coordinates (screenx, screeny) whereby the desired location of the coordinate origin and the orientation of the Y axis are considered during the calculations. The screen coordinates are transferred with the aid of the line list of the drawing routine, which, through the built-in "Cohen- Sutherland clipper” draws only the desired screen area using the border points clipule and cliplri (clip upper left, clip lower right). To create some movement in this house, the rotation origin point or its rotation angle can be changed slightly after each drawing and the whole process can be programmed into a large loop for repeated execution. In case you did not understand a few details, you can relax while typing in the following program listings. You should consider that the material discussed here corresponds to about a half a semester of lectures for upper-class computer science students and therefore requires intense consideration, even with the aid of the additional literature cited in the beginning. Just as in the first small program (random lines) this program is also divided into a link file and main program. The new link file has the name grlinkl. s and was enhanced with the sine and cosine routines, the clip algorithm, the screen switch routine, the matrix operations and the perspective transform routine. The main program housel.s contains the data of the house and the main loop in which the rotation angles of the house are changed in cycle and can be altered by the user. The steps for creating a ready-to-run program are the same as in the third chapter. You need only to replace basicl. s with grlinkl. s and mainl. s with house 1. s in the command sequences. You should start typing in the first program since the following programs build on the first two files. That way you only have to type in the additional subroutines and data sections. The link file grlinkl . s is the same for all following main programs and does not have to be changed. 122 Abacus Software ST 3D Graphics *********************************************************************** * grlinkl.s Graphic Driver Version 4.0 * * The main program must begin with the label " main * *********************************************************************** *********************************************************************** * Global variables in the link files * .globl drawl,sin,sincos,physbase .globl logbase .globl sinx,siny,sinz,cosx,cosy,cosz,wait .globl waitl,drawnl .globl pers,grafhand •globl nummark,xangle,yangle,zangle,numline,datx,daty,datz .globl pointx,pointy .globl pointz,xplot,yplot,xO,yO,zO,zl,linxy,sincos .globl grhandle,global,contrl,intin,intout,ptsin,ptsout .globl addrin,addrout .globl apinit,openwork,clwork,aes,vdi .globl rotate,dist,zobs .globl matrixll,matrixl2,matrixl3 .globl matrix21,matrix22,matrix23 .globl matrix31,matrix32,matrix33 .globl xrotate,yrotate,zrotate,matinit,inkey .globl mouse_on,mouse_off,printf .globl clipxule,clipyule,clipxlri,clipylri .globl filstyle,filindex,filform,filcolor,filmode,yrot .globl lineavar,pageup,pagedown,plotpt 123 ST 3D Graphics Abacus Software ******************************************************************** * Program initialization and storage requirement calculations * ******************************************************************** .text sstart: move.1 a7, a5 * Base page address on the stack move.1 4 (a5),a5 * basepage address = program start - $100 move.1 $c(a5),dO * Program length add. 1 $14(a5),dO * Length of initialized data area add. 1 $lc{a5),d0 * Data area not initialized add. 1 #$1100,dO * 4 K-byte user stack move.1 a5, dl * Start address of the program add. 1 dO, dl * Plus number of occupied bytes = space requirement and. 1 #-2,dl * Even address for stack move.1 dl, a 7 * User stack pointer to last 4K- byte move.1 dO,-(sp) * Length of reserved area move.1 a5,-(sp) * Beginning address of reserved area move.w dO, -(sp) * Dummy-word move.w #$4a,-(sp) * GEM DOS function SETBLOCK trap #1 add. 1 #12,sp * Restore old stack address jsr startl * Check on display address jsr inlinea * Initialize Line-A routines jsr main * Jump to main program (user-created) move.1 #0,- (a7) * End current program trap #1 * Back to Gem desktop *********************************************************************** * Pass upper screen page to video controller * * while drawing the other * *********************************************************************** pageup: move.w #-l,-(a7) move.1 physbase,-(a7) * Page displayed move.1 logbase,- (a7) * Draw on this page move.w #5,- (a7) trap add. 1 #12,a7 rt s 124 Abacus Software ST 3D Graphics *********************************************************************** * Display screen page at lower address, while all drawing * * operations after the call go to the higher display * *********************************************************************** pagedown: move.w #-l,-(a7) move.1 logbase,- (a7) * display logical page move.1 physbase,-<a7) * draw in the other one move. w #5,-(a7) trap Pb add. 1 #12,a7 rts *********************************************************************** * This subroutine calls AES functions, the user must * * save the Registers D0-D2 and A0-A2 before the aes call, * * which are used by VDI and AES * *********************************************************************** aes: move.1 move.w trap rts #aespb,dl #$c8,dO dp * call the AES functions * call the VDI functions * vdi: move.l #vdipb,dl * call the VDI functions move.w #$73,dO trap (^#2^? rts 125 ST 3D Graphics Abacus Software *********************************************************************** * initialize the Line-A functions, pass the address of * * Line-A variable area in AO, which is then stored * * in lineavar * *********************************************************************** .dc.w $a000 move.1 aO,lineavar move. w #0,32(aO) move.w #$ffff,34(aO) move.w #0,36 (aO) move.w rts #1,24 <a0) * announces application * apinit: clr. 1 dO move. 1 dO,aplresv move.1 dO,ap2resv move.1 dO,ap3resv move.1 dO,ap4resv move.w #10,opcode move.w #0,sintin move.w #1,sintout move.w #0,saddrout move.w #0,saddrin jsr aes rts * announces an application A********************************************************************** * Transfers desktop screen handler to caller * *********************************************************************** grafhand: move.w #77,contrl move.w #0,contrl+2 move.w #5,contrl+4 move.w #0,contrl+6 move.w #0,contrl+8 * Transfer screen handler 126 Abacus Software ST 3D Graphics jsr aes move.w intout,grhandle rts * open a workstation * openwork: move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move.w move„w jsr rts #100,opcode #1, dO #0,contrl+2 #11,contrl+6 grhandle,contrl+12 dO,intin dO,intin+2 dO,intin+4 dO,intin+6 dO,intin+8 dO,intin+10 dO,intin+12 dO,intin+14 dO,intin+16 dO,intin+18 #2,intin+20 vdi * opens a workstation * Clear the screen clwork: move.w #3,contrl move.w #0,contrl+2 move.w #1,contrl+6 move.w grhandle,contrl+12 jsr vdi rts * clear screen VDI function 127 ST 3D Graphics Abacus Software * Enable mouse * mouse on: move.w #122,contrl * enable mouse move. w #0,contrl+2 * and control with move. w #1,contrl+6 * operating system move.w grhandle,contrl+12 move.w #0,intin jsr vdi rts * Disable mouse * mouse off: move.w #123,contrl * Disable mouse move. w #0,contrl+2 * and control move.w #0,contrl+6 move.w grhandle,contrl+12 jsr vdi rts * write string on screen * printf: move.1 move.w trap addq.1 rt s aO,-(a7) #9,-(a7) #1 #6,a7 * write a string * whose starting * is in AO, on the * screen. String * must end with a zero. * Determine screen address * start 1: move.w #2,-{a7) trap #14 addq.1 #2,a7 * Determine the screen * address of the system * which computer ? 128 Abacus Software ST 3D Graphics move. 1 sub. 1 move.1 rts dO,physbase * screen start minus 32 K-byte #$8000,dO dO,logbase * equals logical display page ************************************************************************ * Plot routine x-coordinate in d2, y-coordinate in d3 * ************************************************************************ plotpt: movem.1 dO-d2/aO-a2, - (a7) t st. w d2 * X-value less than zero => bmi stop2 tst. w d3 * Y-value less zero bmi stop2 cmp.w #639,d2 * X-value greater than 639? bhi stop2 * Display limit cmp.w #399,d3 * Y-value greater than 399? bhi stop2 move .w d2,ptsin move. w d3,ptsin+2 move. w #1,intin .dc. w $a001 movem. 1 (a7) +,d0-d2/a0-a2 stop2: rts ********************************************************************** * draw-line routine with Cohen-Sutherland clipping. The points are * * passed in d2, d3 (start point) and a2, a3 (end point) ********************************************************************** drawl: movem.1 d0-d7/a0-a6,-(a7) move.w d2,d6 move.w d3,d7 jsr rel_pos move.w dl,codel move.w a2,d6 move.w a3,d7 jsr rel_pos move.w dl,code2 tst. w dl bne testwl * Save registers * Determine position * of start point and * store * Position of second * point and store * if points are not in * drawing area continue 129 ST 3D Graphics Abacus Software * testwl: testw2:, testw3: tst. w codel * test. Otherwise test beq drawit2 * first point. When visible. * draw both points move.w dl, dO * If both points on the same and. w codel,dO * 'page' outside the viewing bne drawend * window, then do not draw. move.w d2,a0 * else store starting points and move.w d3, al * calculate intersecting points move.w a2, a4 move.w a3, a5 tst.w code2 * is point 2 visible ? bne testw2 * if not, find intersection point move.w a2,rightx * if yes, store move. w a3,righty bra testw3 * find left intersect point move.w codel,plcode * right intersect point move.w code2,p2code jsr fndpoint * find intersect point tst. w plcode * if 'intersect point'not bne drawend * visible, then end move.w d2,rightx * if visible, then store move.w d3,righty move.w a4,d2 * and the left intersect point move. w a5, d3 * with switched points move.w aO, a2 * determine with the same routine move.w al, a3 move.w code2,plcode move.w codel,p2code tst. w p2code ★ Point visible? bne testw4 * if not, continue test move.w a2,leftx ★ if yes, store and move. w a3,lefty * connect both visible bra drawitl * points with a line 130 Abacus Software ST 3D Graphics testw4: jsr fndpoint ★ Find intersect point move. w d2,leftx ★ and store. move. w d3,lefty drawitl: move. w leftx,d2 * connect both points with move. w lefty,d3 * a line clr. 1 a2 clr. 1 a3 move. w rightx,a2 move.w righty,a3 drawit2: move.1 lineavar,aO move. w d2,38 <a0) * XI move.w d3,40 (aO) ★ Y1 move.w a2,42(aO) * X2 move.w a3,44(aO) * Y2 .dc .w Sa003 * Draw line drawend: endit: movem.1 (a7)+,d0-d7/a0-a6 ★ Restore registers rts * Return to calling program ********************************************************************** * recognizes the position of a point passed in D6 and D7 relative * * to the clip window defined in the variables clipoli and clipure * ********************************************************************** clr. 1 dl * determines the position move. w d7,dl * of the point passed in sub. w clipyule,dl . * d6 and d7 relative to lsl.l #1, dl * the drawing window move.w d7, dl * defined by clipure sub. w clipylri,dl * and clipoli neg. w dl lsl.l #1, dl move.w d6, dl sub. w clipxlri,dl neg. w dl lsl.l #1, dl move.w d6, dl sub. w clipxule,dl lsl.l #1, dl 131 ST 3D Graphics Abacus Software swap dl rts ********************************************************************** * Finds the intersect point, if present, * * of the the connecting line from Pi to P2 with the clip window * * the points are passed in D2, D3 and A2, A3 as in drawl * ********************************************************************** fndpoint: findwl: move.w d2,d4 * Find the center point of move. w d3,d5 * the line PI P2 add. w a2,d4 * {XI + X2) / 2 ext .1 d4 lsr. 1 #l,d4 add. w a3,d5 * (Y1 + Y2) / 2 ext .1 d5 * = center point of line PI P2 lsr. 1 #l,d5 move.w d4,d6 * Store center point coord. move.w d5,d7 * Y middle jsr rel_pos * where is the intersect point move.w p2code,d6 * Code of center pt. to D6 and.w dl, d6 * are the points on the same bne fother * page outside the screen cmp. w d4,d2 * points coincide ? bne findwl cmp.w d5,d3 beq fendit * if yes => stop cmp. w d4, a2 * Do middle point and second bne findw2 * point match ? cmp. w d5,a3 bne findw2 bra fendit * if yes = stop ? 132 Abacus Software ST 3D Graphics f indw2: move. w d4,d2 * else exchange middle and move. w d5,d3 * first point and start again move. w dl,plcode bra fndpoint fother: cmp - w d4,a2 * middle point and P2 match ? bne fotherl cmp. w d5, a3 beq fendit * if yes, then end fotherl: cmp. w d4,d2 * middle point and PI match ? bne fother2 cmp. w d5,d3 beq fendit * if yes, then end fother2: tst. w plcode ★ is Pl in clip window beq fother3 move.w dl,d7 * if not, and Pl and P2 lie and. w plcode,d7 * both on one side of the bne fexit * Clip-window then none of line is visible fother3: move.w d4,a2 * otherwise take middle point move.w d5,a3 * as new P2 and start again move.w dl,p2code * until the intersect point bra fndpoint * is found fexit: move.w #1,plcode * Inform calling prog, of termination. fendit: rts * either in d2,d3 middle point. or * * in plcode termination notice ****************************************************************** * sine and cosine Function, angle is passed in DO and * the sine and cosine are returned in Dl and D2 * ****************************************************************** tst. w dO * Angle negative, add 360 degrees bpl noaddi add.w #360,dO move.1 #sintab,al ★ Beginning address of sine £able move. 1 d0,d2 * Angle in dO and d2 lsl. w #1, dO * Angle times two as index for access move.w 0 (al,dO.w),dl * sine to dl 133 ST 3D Graphics Abacus Software cmp. w #270,d2 bit plus9 sub. w #270,d2 bra sendsin plus9: add.w #90,d2 sendsin: lsl. w #l,d2 move.w 0 (al,d2.w) ,d2 rt s * Calculate cosine through * displacement of sine values * by 90 degrees * cosine to d2 * * and back to calling program * sine function * * Angle is passed in dO and the sine returned in dl * sin: move. 1 #sintab,al t st • w dO bpl sinl add.w #360,dO sinl: lsl. w #1, dO move.w rts 0(al,dO.w) ************************************************************************ * Initialize the main diagnonal of the result matrix with * * ones which were multiplied by 2 A 14. This subroutine must * * be called at least once before the call by rotate, or the * * result matrix will only consist of zeros. * ★★★★A******************************************************************* move.w #0, dl move.w #16384,d2 * The initial value for move.w d2,matrixll * the main diagonal of move.w dl,matrixl2 * the result matrix move.w dl,matrixl3 * all other elements move.w dl,matrix21 ★ at zero move. w d2,matrix22 move.w dl,matrix23 move.w dl,matrix31 move . w dl,matrix32 134 Abacus Software ST 3D Graphics move.w d2,matrix33 rts ************************************************************************ * Multiplication of the rotation matrix by the rotation * matrix for rotation about the X-axis ************************************************************************ xrotate: move. w xangle,dO jsr sincos move.w dl,sinx move. w d2,cosx move.w dl, d3 move.w d2,d4 move.w matrixll,rotxll move.w matrix21,rotx21 move.w matrix31,rotx31 muls matrixl2,d2 mu Is matrixl3,dl sub. 1 dl, d2 lsl.l #2,d2 swap d2 move. w d2,rotxl2 move. w d3, dl move. w d4,d2 muls matrix22,d2 muls matrix23,dl sub. 1 dl, d2 lsl.l '#2,d2 swap d2 move.w d2,rotx22 move.w d3,dl move.w d4,d2 muls matrix32,d2 muls matrix33,dl sub. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotx32 move.w d3, dl * multiply matrixll-matrix33 * with the rotation matrix for a * rotation about the X-axis * The first column of the matrix * does not change with X rotation 135 ST 3D Graphics Abacus Software move. w d4,d2 mu Is matrixl2,dl mu Is matrixl3,d2 add. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotxl3 move. w d3, dl move. w d4,d2 mu Is matrix22,dl mu Is matrix23,d2 add. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotx23 muls matrix32,d3 mu Is matrix33,d4 add. 1 d3,d4 lsl.l #2,d4 swap d4 move.w d4,rotx33 move. 1 irotxll,al move.1 #matrixll,a2 move.1 #9,d7 * Number of matrix elements subq.1 #1, d7 rotxlopl’: move. w (al)+, (a2) + * Copy result matrix. which dbra d7,rotxlopl * is still in ROTXnn, to MATRIXnn rts *********************************************************************** * multiply the general rotation matrix by the Y-axis * * rotation matrix. Results are stored in the general * * rotation matrix * *********************************************************************** move. w yangle,dO * Angle around which jsr sincos move .w dl,siny move.w d2,cosy move.w dl, d3 * Sine of Y-angle move.w d2,d4 * Cosine of Y-angle 136 Abacus Software ST 3D Graphics mu Is matrixll,d2 mu Is matrixl3,dl add. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotxll move.w d3, dl move.w d4, d2 mu Is matrix21,d2 mu Is matrix23,dl add. 1 dl,d2 lsl.l #2,d2 swap d2 move.w d2,rotx21 move.w d3,dl move.w d4, d2 muls matrix31,d2 muls matrix33,dl add. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotx31 neg.w d3 move. w d3, dl * -siny in the rotation matrix move .w d4,d2 move. w matrixl2,rotxl2 move. w matrix22,rotx22 * The second column move. w matrix32,rotx32 * of the starting muls matrixll,dl * matrix does not muls matrixl3,d2 * change add. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotxl3 move.w d3,dl move.w d4,d2 muls matrix21,dl muls matrix23,d2 add. 1 dl,d2 lsl.l #2,d2 swap d2 move.w d2,rotx23 137 ST 3D Graphics Abacus Software muls matrix31,d3 muls matrix33,d4 add. 1 d3,d4 lsl.l #2,d4 swap d4 move.w d4,rotx33 move.1 #8, d7 move.1 #rotxll,al move.1 #matrixll,a2 yrotlopl: move.w (al)+, <a2)+ dbra d7,yrotlopl rts * Address of result matrix * Address of original matrix * Copy result matrix * to the original matrix * 2-axis - Rotation matrix multiplications * move.w zangle,dO jsr sincos move.w dl,sinz move.w d2,cosz move.w dl,d3 move.w d2, d4 muls matrixll,d2 muls matrixl2,dl sub. 1 dl, d2 lsl.l #2,d2 swap d2 move.w d2,rotxll move.w d3, dl move.w d4,d2 muls matrix21,d2 muls matrix22,dl sub. 1 dl, d2 lsl.l #2,d2 swap d2 move. w d2,rotx21 move. w d3, dl move.w d4,d2 muls matrix31,d2 muls matrix32,dl sub. 1 dl, d2 138 Abacus Software ST 3D Graphics zrotlopl lsl.1 #2,d2 swap d2 move.w d2,rotx31 move.w d3, dl move.w d4, d2 mu Is matrixll,dl mu Is matrixl2,d2 add. 1 dl,d2 lsl.l #2, d2 swap d2 move.w d2,rotxl2 move.w d3, dl move. w d4,d2 mu Is matrix21,dl mu Is matrix22,d2 add. 1 dl, d2 lsl.l #2, d2 swap 42 move. w d2,rotx22 mu Is matrix31,d3 mu Is matrix32,d4 add. 1 d3,d4 lsl.l #2,d4 swap d4 move.w d4,rotx32 move. w matrixl3,rotxl3 * the third column move.w matrix23,rotx23 * remains move.w matrix33,rotx33 * unchanged move . 1 #8,d7 move.1 trotxll,al move.1 #matrixll, a2 : move.w (al)+,(a2)+ * copy to general dbra d7, zrotlopl * rotation matrix rts 139 ST 3D Graphics Abacus Software * Multiply every point whose Array address is in datx etc. * * by previous translation of the coordinate source to * * point [offx,offy,offz], with the general rotation matrix. * * The coordinate source of the result coordinates is then * * moved to point [xoffs,yoffs,zoffs] * rotate: rotatel: move.w nummark,dO * Number of points to be ext. 1 dO * transformed as counter subq.1 #1, dO move.1 datx,al move.1 daty,a2 move.1 datz,a3 move.1 pointx,a4 move.1 pointy,a5 move.1 pointz,a6 move.w (al)+,dl * X-coordinate add.w offx,dl move.w dl, d4 move.w (a2)+,d2 * Y-coordinate add.w offy,d2 * Translation to point [offx,offy,offz] move.w d2,d5 move.w (a3)+,d3 * Z-coordinate add.w offz,d3 move.w d3,d6 muls matrixll,dl mu Is matrix21,d2 muls matrix31,d3 add. 1 dl, d2 add. 1 d2,d3 lsl.l #2,d3 swap d3 add.w xoffs,d3 move. w d3,(a4)+ * rotated X-coordinate move. w d4, dl move.w d5,d2 move.w d6,d3 muls matrixl2,dl Abacus Software ST 3D Graphics mu Is matrix22,d2 mu Is matrix32,d3 add. 1 dl,d2 add. 1 d2,d3 lsl.l #2,d3 swap d3 add. w yoffs,d3 move . w d3,(a5)+ mu Is matrixl3,d4 mu Is matrix23,d5 mu Is matrix33,d6 add. 1 d4,d5 add. 1 d5, d6 lsl.l #2,d6 swap d6 add. w zoffs,d6 move. w d6,(a6)+ dbra rts dO,rotatel * rotated Y-coordinate * rotated Z-coordinate ************************************************************************ * Perspective, calculated from the transformed points in the arrays * * pointx, pointy an d point z the screen coordinates, which * * are then stored in the arrays xplot and^plot . * ************************************************************************ move. 1 pointx,al * Beginning address of move. 1 pointy,a2 * Point arrays move.1 pointz,a3 move.1 xplot,a4 * xplot contains start address of the move.1 yplot,a5 * display coordinate array move.w nummark,dO * Number of points to be transformed ext .1 dO * as counter subq.1 #1, dO move.w (a3)+, d5 ★ z-coordinate of object move.w d5,d6 move. w dist,d4 * Enlargement factor sub. w d5,d4 ★ dist minus Z-coordinate of Obj.coord 141 ST 3D Graphics Abacus Software persl: perendl: ext. 1 d4 lsl.l #8,d4 * times 256 for value fitting move. w zobs,d3 * Projection center Z-coordinates ext. 1 d3 sub. 1 d6,d3 * minus Z-coordinate of object bne persl move. w #0, dl * Catch division by zero addq.1 #2,al * Not really required since addq.1 #2,a2 * computer catches this move.w dl,(a4)+ * with an interrupt move.w dl,(a5)+ bra perendl divs d3,d4 move.w d4,d3 move.w (al)+,dl * X-coordinate of object move.w dl,d2 neg. w dl muls dl,d3 * multiplied by perspective factor lsr.l #8,d3 ★ /256 save value range fitting add.w d3,d2 * add to X-coordinate add.w x0,d2 * add screen offset (center point) move. w d2,(a4)+ * Display X-coordinate move.w (a2)+,dl * Y-coordinates of object move.w dl, d2 neg. w dl muls dl, d4 lsr. 1 #8,d4 * /256 add. w d4,d2 neg. w d2 * Display offset, mirror of Y-axis add.w y0,d2 * Source at [XO,YO] move.w d2,(a5)+ * Display Y-coordinate dbra dO, perlop * All points transformed ? rt s ★ If yes, return 142 Abacus Software ST 3D Graphics •dr***************************************************************** * Draw number of lines from array from lines in linxy * ****************************************************************** drawnl: drlop: move.1 xplot,a4 * Display X-coordinate move.1 yplot,a5 * " Y-coordinate move. w numline,dO * Number of lines ext .1 dO subq.1 #1, dO * as counter move.1 linxy,a6 * Address of line array move.1 (a6) +,dl * first line , (PI,P2) subq.w #l,dl * fit to list structure lsl.w #1, dl * times list element length (2) move.w 0(a4,dl.w),d2 * X-coordinate of second point move.w 0(a5,dl.w),d3 * Y-coordinate of second point swap dl * same procedure for first point subq.w #1, dl lsl.w #1, dl move.w 0(a4,dl.w),a2 * X-coordinate of first point move.w 0(a5,dl.w),a3 * Y-coordinate of first point jsr drawl * draw line from P2 to P2 dbra dO,drlop * All lines drawn ? rts *********************************************************************** * simple counting loop * *********************************************************************** waitl dbra dO,waitl * delay loop, counts dO register rts * down to -1 *********************************************************************** * wait for key press, for Test and Error detection * *********************************************************************** wait: move.w #l,-(a7) * wait for key activation trap #1 * GEM DOS call addq.l #2,a7 rts 143 ST 3D Graphics Abacus Software *********************************************************************** * Key sensing, ASCII code returned in lower byte word of DO * * Scan code in upper sord lower byte of DO * * Returns zero if no input * *********************************************************************** move.w #2,-(a7) * Key sensing, does not move.w #1,- (a7) * wait for a key trap #13 * press addq.1 #4, a7 tst. w dO bpl endkey move.w #7,-{a7) trap #1 addq.1 #2,a7 endkey: rts ** The six following subroutines are only required ** ** for the second main program and do not have to be ** ** entered for linking to the first main program ** filstyle: filindex: move. w #23,contrl * VDI function, set move.w #0,contrl+2 * fill style passed move. w #1,contrl+6 ★ in DO move. w grhandle,contrl+12 move. w dO,intin jsr vdi rts movem.1 dO-d2/aO-a2,- (a7) * set fill pattern move. w #24,contrl * also passed in DO move.w #0,contrl+2 move.w #1,contrl+6 move -w grhandle,contrl+12 move.w dO,intin 144 Abacus Software ST 3D Graphics filcolor: filmode: filform: jsr vdi raovera.1 {a 7)+ ,d0-d2/a0-a2 rts move. w #25,contrl move. w #0,contrl+2 move. w #1,contrl+6 move.w grhandle,contrl+12 move.w #1,intin jsr vdi rts move.w #32,contrl move.w #0,contrl+2 move.w #1,contrl+6 move.w grhandle,contrl+12 move.w dO,intin jsr vdi rts move. w #104, contrl move.w #0, contrl + 2 move.w #1,contrl+6 move.w grhandle,contrl+12 move.w #1,intin jsr vdi rts * set fill color to * one * set write mode * passed in DO * switch on border * around area ********************************************************************* * Rotation of a number of points (nummark) in array datx etc. around* * angle yangle around Y-axis to array pointx = address of array * ********************************************************************* move.w yangle,dO * rotate the definition line jsr sincos * of a rotation body nummark move. w dl,siny * times about the Y-axis move.w d2,cosy * Rotation is done without move.1 datx,al * matrix multiplication. move.1 daty,a2 * but directly, from arrays datx move.1 datz,a3 * in which the address of the definition move.1 pointx,a4 * line was stored into the array 145 ST 3D Graphics Abacus Software move.1 pointy,a5 * whose address is stored move.1 pointz,a6 * in pointx etc. move . w nummark,dO ext. 1 dO * the rotation is about subq.1 #1, dO * angle -y, i.e. from direct: move.w (al)+,dl * positive Y-axis move.w dl,d3 * counterclockwise move.w (a3)+,d2 move.w d2,d4 * z' = x*siny + z*cosy mu Is cosy,d2 lsl.l #2, d2 * retract area extension swap d2 * sine values mu Is siny,dl lsl.l #2, dl swap dl add. w dl,d2 move. w d2,(a6)+ * store z' mu Is siny,d4 * calculate x' lsl.l #2,d4 * x' = x*cosy - z*siny swap d4 neg. w d4 mu Is cosy,d3 lsl.l #2,d3 swap d3 add. w d3,d4 move.w d4, (a4) + * store x' move.w (a2) +, (aS) + * y' = y, since rotation is dbra dO,ylop * around Y-axis rts * Variables for the basic program * • even .data * Sine table starts here sintab: .dc.w .dc. w . dc. w 0,286,572,857,1143,1428,1713,1997,2280 2563,2845,3126,3406,3686,3964,4240,4516 4790,5063,5334,5604,5872,6138,6402,6664 146 Abacus Software ST 3D Graphics .dc.w . dc. w .dc.w .dc.w .dc.w .dc.w .dc.w • dc.w .dc.w .dc.w .dc.w .dc.w .dc.w -dc.w • dc.w • dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc. w .dc.w .dc.w .dc.w .dc.w • dc.w .dc.w .dc.w .dc.w 6924,7182,7438,7692,7943,8192,8438,8682 8923,9162,9397,9630,9860,10087,10311,10531 10749,10963,11174,11381,11585,11786,11982,12176 12365,12551,12733,12911,13085,13255,13421,13583 13741,13894,14044,14189,14330,14466,14598,14726 14849,14962,15082,15191,15296,15396,15491,15582 15668,15749,15826,15897,15964,16026,16083,16135 16182,16225,16262,16294,16322,16344,16362,16374 16382,16384 16382,16374,16362,16344,16322,16294,16262,16225 16182 16135,16083,16026,15964,15897,15826,15749,15668 15582,15491,15396,15296,15191,15082,14962,14849 14726,14598,14466,14330,14189,14044,13894,13741 13583,13421,13255,13085,12911,12733,12551,12365 12176,11982,11786,11585,11381,11174,10963,10749 10531,10311,10087,9860,9630,9397,9162,8923 8682,8438,8192,7943,7692,7438,7182,6924 6664,6402,6138,5872,5604,5334,5063,4790 4516,4240,3964,3686,3406,3126,2845,2563 2280,1997,1713,1428,1143,857,572,286,0 -286,-572,-857,-1143,-1428,-1713,-1997,-2280 -2563,-2845,-3126,-3406,-3686,-3964,-4240,-4516 -4790,-5063,-5334,-5604,-5872,-6138,-6402,-6664 -6924,-7182,-7438,-7692,-7943,-8192,-8438,-8682 -8923,-9162,-9397,-9630,-9860,-10087,-10311,-10531 -10749,-10963,-11174,-11381,-11585,-11786,-11982 -12176 -12365,-12551,-12733,-12911,-13085,-13255,-13421 -13583 -13741,-13894,-14044,-14189,-14330,-14466,-14598 -14726 -14849,-14962,-15082,-15191,-15296,-15396,-15491 -15582 -15668,-15749,-15826,-15897,-15964,-16026,-16083 -16135 -16182,-16225,-16262,-16294,-16322,-16344,-16362 -16374,-16382,-16384 147 ST 3D Graphics Abacus Software .dc.w -16382,-16374,-16362,-16344, -16322,-16294,' -16262 . dc. w -16225,-16182 .dc.w -16135,-16083,-16026,-15964, -15897,-15826, -15749 .dc. w -15668 .dc. w -15582,-15491,-15396,-15296, -15191,-15082, -14962 • dc. w -14849 . dc. w -14726,-14598,-14466,-14330, -14189,-14044, -13894 .dc.w -13741 . dc. w -13583,-13421,-13255,-13085, -12911,-12733, -12551 .dc.w -12365 .dc.w -12176,-11982,-11786,-11585, -11381,-11174, -10963 .dc.w -10749 .dc.w -10531,-10311,-10087,-9860,- 9630,-9397,-9162,-8923 .dc.w -8682,-8438,-8192,-7943,-7692,-7438,-7182, -6924 .dc.w -6664,-6402,-6138,-5872,-5604,-5334,-5063, -4790 .dc.w -4516,-4240,-3964,-3686,-3406,-3126,-2845, -2563 .dc.w -2280,-1997,-1713,-1428,-1143,-857,-572,-286,0 .even • bss xO: • ds. w 1 * Position of the coordinate origin on o > • ds. w 1 * the screen zO: .ds. w 1 zl: .ds. w 1 linxy .ds. 1 1 * This is the address of the line array nurrmark: . ds. w 1 * Number of points numline: -ds. w 1 * Number of lines pointx: . ds. 1 1 * Variables of point arrays for world, pointy: .ds. 1 1 * view, and screen coordinates pointz: • ds. 1 1 xplot .ds.l 1 yplot .ds. 1 1 datx: .ds.l 1 daty: .ds.l 1 datz: .ds.l 1 148 Abacus Software ST 3D Graphics sinx: .ds.w 1 * Temporary storage for sine and sinz: .ds.w 1 * cosine values siny: .ds.w 1 cosx: .ds.w 1 cosz: .ds.w 1 cosy: .ds.w 1 varl: .ds.w 1 * general variables var2: .ds.w 1 var3: .ds.w 1 xangle: .ds.w 1 * Variables for passing angles yangle: . ds . w 1 * to the rotation subroutine zangle: .ds.w 1 physbase: -ds. 1 1 * Address of first screen page logbase: .ds. 1 1 * Address of second screen page contrl: opcode: .ds.w 1 * Arrays for AES and VDI functions * for passing parameters sintin: .ds.w 1 sintout: .ds.w 1 saddrin: .ds.w 1 saddrout: .ds.w 1 .ds.w 6 global: apversion: .ds.w 1 apcount: .ds.w 1 apid: .ds.w 1 apprivate: • ds. 1 1 apptree: .ds. 1 1 aplresv: .ds. 1 1 ap2resv: .ds. 1 1 ap3resv: . ds . 1 1 ap4resv: . ds. 1 1 intin: • ds.w 128 ptsin: .ds.w 256 intout: .ds.w 128 149 ST 3D Graphics Abacus Software ptsout: .ds. w 128 addrin: .ds. w 128 addrout: . ds. w 128 grhandle: .ds. w 1 lineavar: .ds. 1 1 * Starting address of Line-A var . data vdipb: .dc. 1 contrl,intin,ptsin,intout,ptsout aespb: . dc. 1 contrl,global,intin,intout,addrin,addrout leftx: .dc. w 0 lefty: .dc.w 0 rightx: .dc. w 0 righty: . dc. w 0 plcode: .dc.w 0 p2code: .dc. w 0 codel: .dc.w 0 code2: .dc.w 0 mid_code: .dc.w 0 clipxule: .dc.w 0 * Clip window variables clipyule: .dc.w 0 clipxlri: .dc.w 639 clipylri: .dc.w 399 dist: .dc.w 0 zobs: .dc.w 1500 rotxll: .dc.w 16384 * Space here for the result matri rotxl2: .dc.w 0 * matrix multiplication rotxl3: .dc.w 0 rotx21: .dc.w 0 rotx22: .dc.w 16384 rotx23: .dc.w 0 rotx31: .dc.w 0 rotx32: .dc.w 0 rotx33: .dc.w 16384 .bss 150 Abacus Software ST 3D Graphics matrixll: matrixl2: matrixl3: matrix21: matrix22: matrix23: matrix31: matrix32: matrix33: * Space here for the general * rotation matrix Desk File View Options 253882 butes used 1 K PRINTERS X TUTORIAL C FKY COM TTP KLIB PRG OUTPUT PRG SPLIT TTP STANDARD PRT TEXTPRO PRG TUTORIAL TXT XTTUTORI TOC a I nmiffiiM 0 1442236 butes used in 128 itens. BASIC PRG 138944 li-20-Ul BASIC RSC 4648 11-20 i] BASIC HRK 346 11-20- BASIC1 BAR 14801 11-20 OPEN APPLICATION Name: BATCH .TTP Parameters: . aslink grllnkl housel|__- OK I I Cancel 1 IJI IkJUl F:\3DUORK.DIRS 333956 bytes used In ASLINK1 BAT BASIC1 S GRLIHK BAT BRLINK1 0 6RLINK1 S 6R0UND1 S HIDE1 PRG HIDE1 S HOUSE1 PRG H0USE1 S HAIN1 PRG HAIN1 S HAIN1C0 PRG MAIN1C0 S ST 3D Graphics Abacus Software 4.1.1 Explanation of the subroutines used grlinkl.s The transfer of addresses of all data, coordinates, number of corners and lines is not made directly, but through global variables. This increases flexibility and makes it possible to use just one rotation routine. For example, the perspective transformation routine (pers) transforms the data whose beginning addresses are passed in the variables pointx, pointy, pointz and the number of which is passed in the variable nummark, in an array, whose starting address is also passed (xplot, yplot). Because of this it does not matter where data is stored in memory and the amount is irrelevant. For example, the transformation can be carried out for all defined points or only for a few. The brief overview which follows on the subroutines of the link file grlinkl. s should be supplemented with the comments in the program. s st art: Initialize the program. aes: vdi: Call a function from the AES library. Calls a function from the VDI library. ap i n it: Announce an application. openwork: Open a logical display. gra f hand: Returns the number of this logical display. mouse on: Enables the mouse and its controller through the ^ — operating system. mouse_of f: Switches off mouse and controller. sincos: Returns the sine (Dl) and cosine value (D2) of an angle (-360,+360) passed in DO. st art 1: Asks for the display address of the system and recognizes what screen resolution is being used; this serves to determine the two screen pages. 152 Abacus Software ST 3D Graphics clwork: plotpt: drawl: rel_pos: end point: matinit: xrotate: yrotate: zrotate: Mtoate: pers: VDI-Function, clears the current logical display. Plots a point, X-coordinate in D2, Y-coordinate in D3. Draws a line from XI, Y1 to X2,Y2 taking the Clip window specified by the variables clipule, cliplre into account using the line-A routine. Recognizes the area in which the point passed in D6 (X-coord.) and D7 (Y-coord.) lies relative to the clip window. The result is returned in D1 (4-bit code). Finds, if present, an intersection point of the line with the border of the clip window. Initializes the main diagonal of the rotation matrix (matrixll-matrix33) with 16384 which corresponds to a sine value of one. Multiplies the rotation matrix by the matrix for one rotation about the X-axis. Multiplies matrix with the matrix for rotation about the Y-axis. Same for Z-axis. This is the general rotation routine. Here every point from the point array (passed in point x etc.) is rotated around the angles xw, yw, zw, and then is moved to point [xoffs, yoffs, zoffs] after a preliminary displacement of the coordinate origin to point [of fx, of fy, of f z], Calculates the perspective screen coordinates and stores them at addresses passed in xplot, yplot. 153 ST 3D Graphics Abacus Software symbol: Connects the points in the screen coordinate array with lines. The address of the line array is in linxy, and the number of lines in numlin. — pagedown: Turns on the logical screen page. After the call drawing is done on the other page. page up: Turns on the physical (higher) display page. Subsequent drawing is done on the logical page (toggle). waitl: A timer loop which only counts the DO -register down to -1. wait: Waits for a key press and then returns. inkey: Senses the keyboard without waiting. The ASCII and key codes are returned in register D 0. printf: Writes a string on the display which must be terminated with a zero. The address is passed in AO. This routine, and the five following routines are not used by the first main program. It rotates a number of points around the Y-axis directly and without use of matrix multiplication. The VDI function sets the fill style which is passed in DO (0=no fill, l=fill with color, 2=fill with dots, 3=shade, 4=user-defined fill pattern). f i 1 index: Sets the various fill patterns according to style f ilcolor: Determines the fill color (for monochrome display only black or white, l=black). f ilmode: Sets the write mode, 1 = replace. filform: Subsequent filled surfaces will be surrounded with a border after calling this routine. yrot: 5 / filstyle: \ 154 Abacus Software ST 3D Graphics ********************************************************************** * housel.s 14.1.1986 * * Display a wire-model house Uwe Braun 1985 Version 1.1 * * * ********************************************************************** .globl main,xoffs,yoffs,zoffs,offx,offy,offz .globl viewx,viewy,viewz .globl wlinxy,setrotdp,inp_chan,pointrot .text main: jsr apinit * Announce program jsr grafhand * Get screen handler jsr openwork ★ Announce screen jsr mouse off * Turn off mouse jsr getreso * which monitor is connected ? jsr setcocli * Set clip window jsr makewrld ★ Create the world system jsr worldset * Pass the world parameters jsr setrotdp ★ initialize obs. ref. point jsr clwork ★ erase both screen pages jsr ('•"pagedown^ 5 * Display logical screen page , j jsr clwork . Input and change parameters' ^ * jsr inp chan ★ mainlopl: ]sr jsr jsr jsr jsr jsr jsr jsr jsr jsr jsr * rotate around obs. ref. point * perspective transformation * Draw lines in linxy array inp_chan clwork * Display physical screen page * Input new parameters * erase V^logical,; screen page pointrot * Rotate around Rot. ref. point pers * Transform, of new points d rawnl _* draw in l ogi cal page, t hen _ pagedowp !) * display this logical page inp chan- * Input and change parameters 155 ST 3D Graphics Abacus Software jsr clwork * erase physical page jmp mainlopl * to main loop move. 1 physbase,logbase jsr (pageup) * switch to normal display page rts back to linkfile, and end ******************************************************************** * Remove all accumulated characters in the keyboard buffer " ******************************************************************** clearbuf: clearend: move.w #$b,-(a7) trap #1 addq.1 #2,a7 tst. w dO beq clearend move.w #1, -(a7) trap #1 addq.1 #2,a7 bra clearbuf rts * Gemdos funct. Character in buffer? * If yes, get character * If no, terminate * Gemdos funct.CONIN * repeat until all characters are * removed from the buffer ********************************************************************** * Change observation parameters with keyboard sensing * * Angle increments, location of the projection plane, etc. ********************************************************************** inp chan: jsr inkey cmp .b #' D',dO bne inpwait jsr scrdmp inpwait: swap dO cmp .b #$4d,dO bne inpl addq.w #1,ywplus bra inpendl inpl: cmp -b #$4b,dO bne inp2 * Read keyboard, code in DO * shift D = print * make hardcopy * test DO, if * Cursor-right * if yes, add one to Y-angle * increment and continue * Cursor-left, if yes, then * subtract one from Y-angle 156 Abacus Software ST 3D Graphics subq.w #1, ywplus bra inpendl inp2: cmp .b #$50,dO bne inp3 addq. w #1,xwplus bra inpendl inp3: cmp. b #$48,dO bne inp3a subq.w #1,xwplus bra inpendl inp3a: cmp .b #$61,dO bne inp3b subq.w #1,zwplus bra inpendl inp3b: cmp.b #$62,dO bne inp4 addq.w #1,zwplus bra inpendl inp4: cmp.b #$4e,dO bne inp5 sub. w #25,dist bra inpendl inp5: cmp.b #$4a,dO bne inp6 add.w #25,dist bra inpendl inp6: cmp.b #$66,dO bne inp7 sub. w #15,rotdpz bra inpendl inp7: cmp.b #$65,dO bne inplO add. w #15,rotdpz bra inpendl * increment * Cursor-down, if yes * then add one to X-angle increment * Cursor-up * subtract one * Undo key * Help key * plus key on numerical keypad * if yes, subtract 25 from location * Projection plane (Z-coordinate) \ * minus key on the numerical keypad * * if yes, add 25 * astersisk key on numerical keypad * if yes, subtract 15 from rotation * point Z-coordinate * Make changes * Division key on num.keypad * add 15 157 ST 3D Graphics Abacus Software inplO: cmp .b #$44,dO bne inpendl addq. 1 #4,a7 bra mainend inpendl: move. w hyangle,dl add. w ywplus,dl cmp. w #360,dl bge inpend2 cmp. w #-360,dl ble inpend3 bra inpend4 inpend2: sub. w #360,dl bra inpend4 inpend3: add.w #360,dl inpend4: move.w dl,hyangle move . w hxangle,dl add.w xwplus,dl cmp.w #360,dl bge inpend5 cmp.w #-360,dl ble inpend6 bra inpend7 inpend5: sub. w #360,dl bra inpend7 inpend6: add.w #360,dl inpend7: move.w dl,hxangle move. w hzangle,dl add.w zwplus,dl cmp. w #360,dl bge inpend8 cmp. w #-360,dl ble inpend9 bra inpendlO inpend8: sub. w #360,dl bra inpendlO inpend9: add. w #360,dl * F10 activated ? * if yes, jump to * program end * Rotation angle about Y-axis * add increment * if larger than 360, then * subtract 360 * is smaller than 360, then * add 360 * proceed in the same manner * with the rotation angle about * the X-axis * store angle 158 Abacus Software ST 3D Graphics inpendlO: move.w dl,hzangle rts * Initialize the rotation reference point to [0,0,0] * move . w #0, dl * set the start-rotation- move . w dl,rotdpx * datum-point move. w dl,rotdpy move.w dl,rotdpz move . w #0,hyangle * Start-rotation angle move. w #0,hzangle move.w #0,hxangle rts * Rotation around one point, the rotation reference point * pointrot: move.w hxangle,xangle move.w hyangle,yangle move.w hzangle,zangle move . w rotdpx,dO move.w rotdpy,dl move.w rotdpz,d2 move. w dO,xoffs move.w dl,yoffs move.w d2,zoffs neg. w dO neg. w dl neg. w d2 move.w dO,offx * move.w dl, of fy move.w d2,offz jsr matinit * jsr zrotate * jsr yrotate * jsr xrotate * jsr rotate * rts * rotate the world around the angle * hxangle, hyangle, hzangle about the * rotation reference point * add for back transformation. subtract for transformation. Matrix initialization first rotate around Z-axis rotate 'matrix' around Y-axis then rotate around X-axis Multiply points with the matrix. 159 ST 3D Graphics Abacus Software * Creation of the world system from the object data * makewrld: move.1 thousdatx, al * create the world system by move.1 Ihousdaty,a2 move.1 #housdatz,a3 move.1 #worldx,a4 move.1 #worldy,a5 move.1 #worldz,a6 move.w hnummark,dO ext .1 dO subq.1 #1, dO makewll: move.w (al) +, (a4) + * copying the house data into move.w (a2)+, (a5) + * world data move. w (a3)+, (a6) + dbra dO,makewll move . w hnumline,dO ext. 1 dO subq.1 U,dO move. 1 thouslin,al move . 1 dwlinxy,a2 makewl2: move. 1 (al) +, (a2) + dbra d0,makewl2 rts ********************************** * Pass the world parameters to the ********************************** worldset: move.1 #worldx,datx * move.1 #worldy,daty * move.1 #worldz,datz move.1 #viewx,pointx move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,linxy move.w picturex,xO move. w picturey,yO move.w proz,zobs move.w rlzl,dist move.1 #screenx,xplot ********************************** link file variables * ********************************** Pass variables for the rotation routine 160 Abacus Software ST 3D Graphics move.1 Iscreeny,yplot move.w hnumline,numline move.w hnummark,nummark rts ********************************************************************* * sense current display resolution and set coordinate origin of the * * screen system to the center of the screen * ********************************************************************* getreso: move.w #4,-(a7) trap #14 addq.1 #2,a7 cmp.w #2, dO bne getrl move. w #320,picturex * for monochrome monitor move.w #200,picturey bra getrend getrl: cmp.w #1, dO bne getr2 move. w #320,picturex * medium resolution (640*200 move.w #100,picturey bra getrend getr2: move.w #160,picturex * low resolultion (320*200) move.w #100,picturey getrend: rts ********************************************************************** * Hardcopy of the display after activating Shift d on keyboard * ********************************************************************** scrdmp: move.w #20,-(a7) trap #14 addq.l #2,a7 jsr clearbuf * prevent another hardcopy rts 161 ST 3D Graphics Abacus Software ********************************************************************* * Sets the limit of the display window for the draw-line algorithm * * built into the Cohen-Sutherland clip algorithm * * The limits are freely selectable by the user, making the draw- 4 * line algorithm very flexible. ********************************************************************* move.w #0, clipxule * Clip left X-Coord. move.w #0,clipyule ★ ” Y-Coord move. w picturex,dl lsl. w #1, dl * times two subq.w #1, dl * minus one equal move.w dl,clipxlri * 639 for monochrom move.w picturey,dl lsl. w #1, dl * times two minus one equal subq.w #1, dl * 399 for monochrom move.w dl,clipylri ★ Clip right Y-Coord rts .even * Here begins the variable area for the program module * * * * * * Definition of the house * * * . data housdatx: .dc.w ,dc. w -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 30,30,30,30,30,30,30,30,30,30,30,30 housdaty: .dc.w .dc.w .dc.w 30,30,-30,-30,30,30,-30,-30,70,70,-30,0,0,-30 20 , 20 , 0 , 0 , 20 , 20 , 0,0 -10,-10,-30,-30 162 Abacus Software ST 3D Graphics housdatz: .dc.w .dc.w .dc.w houslin: .dc.w .dc.w .dc.w .dc.w hnummark: .dc.w hnumline: .dc.w hxangle: .dc.w hyangle: .dc.w hzangle: .dc.w xwplus: .dc.w ywplus: .dc.w zwplus: .dc.w picturex: • dc.w picturey: .dc.w rotdpx: .dc.w rotdpy: .dc.w rotdpz: .dc.w rlzl: .dc.w normz: . dc. w .bss plusrot: . ds. 1 first: .ds. 1 second: . ds . w deltal: .ds. w .data 60,60,60,60,-60,-60,-60,-60,60,-60,60,60,60,60 40,10,10,40,-10,-40,-40,-10 0 ,- 20 ,- 20,0 1,2,2,3,3, 4,4,1,2,5,5,8,8,3,8,7,7,6,6,5,6,1,7,4 9, 10, 1, 9, 9, 2,5,10, 6,10, 11,12, 12, 13, 13,14 15,16,16,17,17,18,18,15,19,20,20,21,21,22,22,19 23,24,24,25,25,26,26,23 26 * Number of corner points of the house 32 * Number of lines of the house 0 * Rotation angle of the house around X-axis 0 * " " " Y-axis 0 * " " " Z-axis 0 * Angle increment around the X-axis 0 * Angle increment around the Y-axis 0 * Angle increment around the Z-axis 320 * Definition of zero point of display 200 * here it is in the display center 0 * Rotation datum point 0 0 0 1500 1 1 1 1 163 ST 3D Graphics Abacus Software flag: . dc .b . even -bss 1 diffz: .ds.w 1 dx: .ds .w 1 dy: -ds.w 1 dz: .ds.w 1 worldx: .ds.w 1600 * World coordinate array worldy: -ds. w 1600 worldz: .ds.w 1600 viewx: .ds.w 1600 * View coordinate array viewy: .ds.w 1600 viewz: .ds.w 1600 screenx: .ds.w 1600 * Display coordinate array screeny: .ds.w 1600 wlinxy: .ds. 1 3200 * Line array .data prox: .dc. w 0 * Coordinates of the Projection- proy: .dc. w 0 * center, on the positive proz: . dc. w 1500 * Z-axis . data of fx: .dc. w 0 * Transformation during Rotation of fy: . dc. w 0 * to point [offx,offy,offz] of fz: . dc. w 0 xoffs: .dc.w 0 * Back transformation to Point yoffs: .dc. w 0 * [xof yof f s, zof fs] zoffs: .dc.w 0 7 .bss loopc: .ds. 1 1 • end 164 Abacus Software ST 3D Graphics 4.1.2 Description of the Subroutines of the first Main program: main: This is the entry point to the program module. The program announces itself and initializes the AES and VDI functions and senses the current screen resolution. The window size and the screen are determined from the resolution. The program section between the labels mainopl: and mainend: is the main loop, which is repeated until the F10 key is pressed. makewrld: Creates a world in the world coordinate system by simple copying of the house data into the world system. These are the coordinates of the house (housdatx, housdaty, housdatz) in the world coordinate system (wrldx, wrldy, wrldz), the lines of the house in houslin in the world line storage area (wlinxy), the number of comer points the house (hnummark) in the total- number variable of the world system (nummark) and finally the number of house lines (hnumline) in numline. This subroutine need only be called once unless you want to add objects to the world system which we will do in a later program. wrldset: After creating the world system the array addresses (wrldx etc.) must be passed to the global variables of the rotation subroutine (datx etc.). Furthermore the coordinate origin of the display is determined in the Variables XO and YO, and the presets for the perspective parameters (zobs, dist). setrotdp: Initializes the rotation reference point to [0,0,0] and the rotation angles to 0 degrees. 165 ST 3D Graphics Abacus Software pointrot: inp chan: getreso: scrdmp: setcocli: clearbuf: This subroutines provides the rotation routine with the current data and then performs the rotation around the point [rotdpx, rot dp, rotdpz] of all three axes with a call to the proper routines of the link file, in the sequence Z-axis, Y-axis, X-axis. A change in the sequence also changes the results. Input and change the parameters, rotation angle, rotation reference point and position of the projec¬ tion plane. Checks the current display resolution and from this determines the data for the screen center and the clip window, which in this case is the whole visible display. Hardcopy routine, is called form inp_chan by pressing shift ’D’ and replaces the key combination Altemate/Help, which the operating system uses to make a hardcopy of the screen. Since in this program the displayed page is never the same as the page in which the drawing occurs, a hardcopy through Altemate/Help would not correspond to the displayed picture but would print the picture under construction or the just-erased display. The trick is to call the scrdmp routine before the displayed page is erased. Set the clip-window for the Cohen-Sutherland clip algorithm on the whole display, 0,0 to 639,399 hi¬ res, 639,199 med-res, or 319,199 lo-res. Remove characters that may be in the keyboard buffer. Is used only by the hardcopy routine, since several hardcopies could otherwise be made in succession (Key repeat). 166 Abacus Software ST 3D Graphics 4.1.3 General comments on the program The specific explanations of the variables can be found in the remarks in the program listing. In each iteration of the main loop the program adds an angle increment (xwplus, ywplus, zwplus) to the rotation angle (hxangle, hyangle, hzangle) of the house. The input routine changes the angle increments which causes the house to rotate faster on the screen, though this is really an optical illusion. The end points of the house have to travel a longer distance between each drawing operation, which causes this effect. The cursor keys, the <Help> and <Undo> keys control the rotation, the V and ’-’keys change the display size by moving the projection plane, and the ’/’ and ’*’ keys move the rotation reference point on the Z-axis. Pressing of the shift and ’D’ keys at the same time produces hardcopy if a printer is attached. The best thing to do is to try out the various changes possible, preferably by changing the constants in the listing. You can, for example move the rotation reference point on the X and Y axis, or the variable proz, which changes the position of the projection center. The closer you move the projection center in the direction of the house, the greater the perspective distortion. You should also define an object yourself, and you should start with a simple object, like a pyramid. You only have to enter the points of the pyramid (in a pyramid with a quadratic base there are five) in place of the house coordinates in the arrays (housdatx etc.). Furthermore, the number of points (5) must be entered in houslin in hnummark, the number of lines (8) in hnumline and then the information regarding which points are connected by lines. You only have to change the storage area and you can represent any defined object with the same program. Here I want to provide some additional information about the storage space required. The arrays (wrldx etc., viewx, screenx, wlinxy) are already dimensioned quite generously for future expansion. You can define objects with 1600 comers and connect these comers with 3200 lines. About 40 KByte of storage space is needed for this array dimensioning. Even though 1600 comers appear to be sufficient at first glance, we shall reach this number in the next chapter without too much effort. But first of all stop for a while and play around with this program. You can also add a window on the other side of the house by simply entering the new coordinates. 167 LIS projection centers Abacus Software ST 3D Graphics 4.2 Generation techniques for creating rotating objects c \ 1'4,-fc, ? If you have experimented with the construction of new objects, you probably also noticed the considerable effort involved in construction, especially for regularly-formed bodies with many comers. Imagine if you had to input the end points of the ball approximated by polygons (See figure 4.2.1). Figure 4.2.1: Hardcopy of the rotation ball The drudgery of input can be performed by the computer for all axis- symmetrical objects. As an example, consider the ’’chess piece’’ from Figure 4.2.2. This figure can be created by rotating a line (the definition line) around any axis, in this case the Y-axis. The programmer must 169 ST 3D Graphics Abacus Software define the one line and indicate how many times it should be rotated. You can follow the construction of the figure easily on the following hardcopies. The rotation number must be a division of 360 for programming reasons or a portion of the figure will be missing. From two to four to three hundred sixty rotations are available. More than 180 just produces a heap of points on the display (the screen resolution is too low). Now the space requirement will become obvious. If you rotate the 12 points 360 times it results in 4,332 points not to mention the 8,291 lines created by the rotation. The number of points is calculated as follows: nummark: =numpt* (rotations + 1). The lines include the connecting lines of the points in the rotating definition line as well as the horizontal connecting lines of the points in the rotation line. The routines for the creation of the rotation body are contained in the listing of the file rotate 1. s. The rotation body is described by a line, i.e. a number of points (rlnumpt), whose coordinates are in rlxdat, rlydat, r 1 zdat and the number of rotations about the Y-axis which this line should perform. The different bodies are created by varying the number of rotations. The maximum number of rotations in our case is 120, which is predetermined by the dimensioning of the array to 1600 etc. and of course could be changed. The number of points of the rotation body is contained in the variable rlnumpt. The link file remains the same as in the first program. You only have to assemble the first file and link it to the link file: as link grlinkl rotatel. Desk File View Options A:\ 253882 butes used i 8 PRINTERS 8 TUTORIAL C FKY C0NV TIP NL1B PRC OUTPUT PRC SPLIT TTP STftNDRRD PRT TEXTPRO PRC TUTORIRL TXT XTTUTORI TOC D:\ K 1442236 butes used in 123 ltens. BASIC PR6 BfiSIC RSC BASIC HRK BASIC1 BAK 138944 11-28 4648 11-28: 346 11-2 14801 11-2 F:\3DU0RK.DIRS 333556 butes used tn OPEN APPLICATION Nanei BATCH Parameters: aslink grlinkl rotatei[ • TTP I OK I ^ | Cancel I a* a* a* a? WBSTSE23M I'-H'Ji i"?u■ iinn i : *tw ttnm H0USE1 PRB NQUSE1 S MAIN1 PRG MAIN1 S MAIN1C0 PRC HAIN1C0 HENU1 MENU1 MULTIi MUL Til PAINT1 PAINT1 R0TATE1 PRG R0TATE1 S S PRG S PRG S PRG S 170 Abacus Software ST 3D ST 3D Graphics Abacus Software ********************************************************************** * rotatel.s 16.1.1986 * * Creation of rotation bodies Uwe Braun 1985 Version 2.0 * .text .globl main,xoffs,yoffs,zoffs,offx,offy, offz -globl viewx,viewy,viewz .globl wlinxy,mouse off,setrotdp,inp chan,pointrot main: jsr apinit * jsr grafhand * jsr openwork it jsr mouse_off * jsr getreso * jsr setcocli * jsr makerotl jsr makewrld * jsr wrld2set * jsr setrotdp * jsr clwork jsr pagedown * jsr clwork jsr inp_chan * mainlopl: jsr pointrot * jsr pers * jsr drawnl jsr pageup * jsr inp_chan * jsr clwork * jsr pointrot * jsr pers * jsr drawnl jsr pagedown * jsr inp_chan * Announce program Get screen handle Display Turn off mouse Which monitor is connected ? Set clip window Create world system Pass world parameters initialize observation ref. point Display logical screen page Input and change parameters rotate around observation ref. point Perspective transformation Display physical page Input new parameters Erase logical page Rotate around rotation ref. point Transform, new points Display this logical page Input and change 172 Abacus Software ST 3D Graphics mainend: jsr clwork * clear physical page jmp mainlopl * to main loop move.1 physbase,logba: se jsr pageup * switch to normal display page rts * back to link file, and end * * * **************** remove all characters from the keyboard buffer ******************************************************************* clearbuf: move.w #$b,-(a7) trap #1 addq.1 #2,a7 tst -w dO beq clearend move.w #1,- (a7) trap #1 addq.1 #2,a7 bra clearbuf clearend: rts * Gemdos funct. char in buffer? * if yes, get character * if no, terminate * Gemdos funct. CONIN * repeat until all characters * are removed from the buffer * Create the rotation body rl makerotl: jsr jsr jsr rt s rlset rotstart rotlin * Create the rotation body * first the coordinates, * then the lines 173 ST 3D Graphics Abacus Software *******************it** + -k1t*-k-k**ic1t1r*1i1iii*1t1r1tl!*1cirit**ir1'1c1r1r***1i1t**i'*iiitit1i1r1rit * Input and change observation parameters * * the angles hxangle,hyangle,hzangle, are rotation angles of * * world system * inp chan: jsr inkey * cmp.b #'D',dO bne inpwait jsr scrdmp * inpwait: swap dO * cmp.b #$4d,dO * bne inpl addq. w #1,ywplus * bra inpendl * inpl: cmp.b #$4b,dO * bne inp2 * subq.w #1,ywplus * bra inpendl inp2: cmp.b #$50,dO * bne inp3 addq.w #1,xwplus * bra inpendl inp3: cmp.b #$48,dO * bne inp3a subq. w #1,xwplus * bra inpendl inp3a: cmp.b #$61,dO ★ bne inp3b subq.w #1,zwplus * bra inpendl inp3b: cmp.b #$62,dO * bne inp4 addq. w #1, zwplus * bra inpendl Sense keyboard, code in make hardcopy test DO if Cursor-right if yes, add one to Y-angle increment and continue Cursor-left, if yes subtract one from Y-angle increment Cursor-down, if yes add one to X-angle increment Cursor-up subtract one Undo-key lower Z-increment Help-key add to Z-increment 174 Abacus Software ST 3D Graphics inp4: cmp .b #$4e,dO * plus key on keypad bne inp5 * if yes, subtract 25 from sub. w #25,dist * position of projection bra inpendl * plane (Z-coordinate) inp5: cmp.b #$4a,d0 * minus key on keypad bne inp6 * add.w #25,dist * if yes, add 25 bra inpendl inp6: cmp.b #$66,dO * times-key on the keypad bne inp7 * if yes, then subtract 15 sub. w #15,rotdpz * from the rotation ref. point Z-coord. bra inpendl * make changes inp7: cmp.b #$65,dO * division-key on keypad bne inplO add.w #15, rotdpz * add 15 bra inpendl inplO: cmp.b #$44,dO * F10 activated ? bne inpendl addq.1 #4,a? * if yes, jump to bra mainend * Program end inpendl: move. w hyangle,dl * rotation angle, Y-axis add.w ywplus,dl * add increment cmp.w #360,dl * if larger than 360, then subtract 360 bge inpend2 cmp. w #-360,dl * if smaller than 360, ble inpend3 * add 360 bra inpend4 inpend2: sub. w #360,dl bra inpend4 inpend3: add. w #360,dl inpend4: move. w dl,hyangle move. w hxangle,dl * proceeed in the same add. w xwplus,dl * manner with rotation cmp.w #360,dl * angle, X-axis bge inpend5 cmp. w #-360,dl 175 ST 3D Graphics Abacus Software ble inpend6 bra inpend7 inpend5: sub. w #360,dl bra inpend7 inpend6: add. w #360,dl inpend7: move. w dl,hxangle move. w hzangle,dl add. w zwplus,dl cmp. w #360,dl bge inpend8 cmp. w #-360,dl ble inpend9 bra inpendlO inpend8: sub. w #360,dl bra inpendlO inpend9: add.w #360,dl inpendlO: move.w rts dl,hzangle * Initialize the rotation reference point to [0,0,0] * move.w #0, dl * set the start-rotation move.w dl,rotdpx * reference point move.w dl,rotdpy move. w dl,rotdpz move. w #0,hyangle * Start rotation angle move.w #0,hzangle move. w #0,hxangle rts ****************************************************************** * Rotation of the total world system around the rotation * * reference point * ****************************************************************** pointrot: move.w move.w move.w move.w hxangle,xangle * rotate the world around hyangle,yangle hzangle,zangle rotdpx,d0 * the rotation reference point 176 Abacus Software ST 3D Graphics ★ makewrld: makewll: jnakewl2 : move.w rotdpy,dl move . w rotdpz,d2 move. w dO,xoffs * add for inverse transformation move . w dl,yoffs move.w d2,zof fs neg. w dO neg. w dl neg. w d2 move. w dO,offx * subtract for transformation move. w dl, offy move. w d2,offz jsr matinit * matrix initialization jsr zrotate * rotate around Z-axis first jsr yrotate * rotate 'matrix' around Y-axis jsr xrotate * then rotate around X-axis jsr rotate * multiply points with the rts * matrix. The Z-axis is not taken into * account move. 1 #rldatx,al * create the world system move . 1 #rldaty,a2 * by copying data of rotation body move.1 ffrldatz, a3 * into world system move.1 #worldx,a4 move. 1 #worldy,a5 move. 1 #worldz,a6 move. w rlnummark,dO * number of corners repeated ext .1 dO subq. 1 #1, dO move . w (al)+, (a4) + * Copy coordinates move . w (a2)+, (a5) + * Y-coords. move. w (a3)+, (a6) + * Z-coords. dbra dO,makewll move.w rlnumline,dO * Copy the line arrays ext. 1 dO * of the rotation body subq.1 #1, dO * into the world system move.1 #rllin,al * Number of lines as counter move.1 #wlinxy,a2 move.1 (al)+,(a2)+ * copy lines dbra dO,makew!2 rt s 177 ST 3D Graphics Abacus Software *********************************************************************** * Pass world parameters to variables of link files * *********************************************************************** worldset: move.l #worldx,datx move.1 fworldy,daty move.1 #worldz,datz move.1 #viewx,pointx move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,linxy move.w picturex,xO move.w picturey,yO move.w proz,temp move.w rlzl,dist move.1 iscreenx,xplot move.1 iscreeny,yplot move.w hnumline,numline move.w hnummark,nummark rts * Passing house variables * for the rotation routine * and the global subroutine * of the link module * Projection center Z-coordinate * Location of projection plane on * the Z-axis * Number of house lines * Number of house corners *********************************************************************** * Creation of rotation body in the array, the address of which * * is passed in the variables rotdatx, rotdaty, rotdatz * *********************************************************************** rlset: move. 1 #rlxdat,rotxdat ★ Transmit move. 1 #rlydat,rotydat * parameters of this move.1 #rlzdat,rotzdat ★ rotation body to move.1 irldatx,rotdatx ★ the routine for move.1 ffrldaty, rotdaty * creation of the move.1 #rldatz,rotdatz * rotation body move.1 rotdatx,datx move. 1 rotdaty,daty move. 1 rotdatz,datz move. w rlnumro,numro * Number of desired move.w rlnumpt,numpt * rotations. Number move.l #rllin,linxy * of points in def.line rts ★ Address of line array 178 Abacus Software ST 3D Graphics rotstart: rloopl: move.w numpt,dO lsl. w #1, dO ext. 1 dO move.1 dO,plusrot move.w numpt,nummark move. 1 rotdatx,pointx move.1 rotdaty,pointy move.1 rotdatz,pointz move.w #0,yangle move.w #360,dO divs numro,dO move.w dO, plusagle move. w numro,dO ext. 1 dO move.1 dO,loopc move.1 rotxdat,datx move.1 rotydat,daty move.1 rotzdat,datz jsr yrot move. 1 pointx,dl add. 1 plusrot,dl move. 1 dl,pointx move. 1 pointy,dl add. 1 plusrot,dl move. 1 dl,pointy move. 1 pointz,dl add. 1 plusrot,dl move. 1 dl,pointz move.w yangle,d7 add. w plusagle,d7 move. w d7,yangle move. 1 loopc,dO dbra dO,rloopl move. w rlnumro,numro move.w rlnumpt,numpt rts * Rotate def line * numro+1 about Y-axis * Pass data array * to subroutine yrot * 360 / numro = angle increment * per rotation * numro +1 times * as loop counter * for passing to yrot * rotate * add offset to * address * Add angle increment * to rotation angle * and rotate line * again until all * end points are generated. * store for following * routines for line generation 179 ST 3D Graphics Abacus Software rotlin: rotlopl: rotlop2: rotlop3 : * Create the line array of the move.w #1, d7 * rotation body move.w numro,d4 * Number of rotations repeated ext. 1 d4 subq. 1 #1, d4 move.w numpt,dl subq.w *1, dl lsl. w #2, dl ext. 1 dl move.1 dl,plusrot move.w numpt,d5 * Number of points - ext. 1 d5 * repeat once subq.1 #2,d5 move.1 linxy,al * Lines created stored move.w d7,d6 * here move.w d6, (al) + * The first line goes from addq.w #1, dG * point one to point two move.w d6, (al) + * (1,2) then (2,3) etc. dbra d5,rotlop2 move. 1 linxy,dl * generate cross connections add. 1 plusrot,dl * of individual lines move.1 dl,linxy move.w numpt,dO add.w d0,d7 dbra d4,rotlopl move. w numpt,d7 move.w d7,deltal lsl. w #2,d7 ext. 1 dl move.1 d7,plusrot move.w #l,d6 move. w numpt,dO ext. 1 dO subq.1 #1, dO move.w numro,dl ext. 1 dl subq.1 #1, dl move.w d6,d5 180 Abacus Software ST 3D Graphics move.w d5, (al) + add. w deltal,d5 move.w d5,(al)+ dbra dl,rotlop4 add. w #l,d6 dbra dO,rotlop3 move. w numro,dl add. w #1, dl mul s nummark,dl move.w dl,rlnummark move.w numpt,dl muls numro,dl move . w numpt,d2 subq.w #1, d2 muls numro,d2 add. w dl, d2 move.w rts d2,rlnumline * Store total number of * corners created * Total of lines created * Pass parameters of the world system to variables * * of the link file for the rotation body * wrld2set: move.1 iworldx,datx * Pass parameter of move.1 fworldy,daty * rotation body ' to the move.1 #worldz,datz * subroutines in the link move.1 #viewx,pointx * module move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,1inxy move.w picturex,xO move.w picturey,yO move. w proz,temp move.w rlzl,dist move.1 Iscreenx,xplot move.1 #screeny,yplot 181 ST 3D Graphics Abacus Software move.w rlnumline,numline * Number of lines move.w rlnummark,nummark * Number of corners rts a********************************************************************** * Sense current display resolution and set the coordinate * * origin of the screen system to the screen center * *********************************************************************** getreso: move. w #4,-(a7) trap #14 addq. 1 #2,a7 cmp. w #2, dO bne getrl move.w #320,picturex * monochrome monitor move. w #200,picturey bra getrend getrl: cmp. w #1, dO bne getr2 move. w #320,picturex * medium resolution (640*200 move.w #100,picturey bra getrend getr2: move.w #160,picturex * low resolution (320*200) move.w #100,picturey getrend: rts * Hardcopy after inp_chan call * scrdmp: move. w trap addq. 1 jsr rts #20,-(a7) #14 #2, a7 clearbuf 182 Abacus Software ST 3D Graphics ********************************************************************* * Set the limit of the window for the Cohen-Sutherland * * clip algorithm built into the draw-line algorithm * * The user can choose the limits freely, which makes the * * draw-line algorithm very flexible. * ********************************************************************* move.w #0,clipxule move.w #0, clipyule move.w picturex,dl lsl .w #1, dl ★ times two subq.w #1, dl A minus one equals move.w dl,clipxlri * 639 for monochrom move.w picturey,dl lsl .w #1, dl ★ times two minus one subq.w #1, dl ★ equals 399 for monochrom move.w dl,clipylri rts . even * Begin variable area for Program module * * * * Data area for the rotation body * .bss * Space for the variables numro: .ds.w 1 numpt: .ds.w 1 worldfla: .ds.l 1 rotxdat: .ds.l 1 rotydat: .ds.l 1 rotzdat: .ds.l 1 ST 3D Graphics Abacus Software rotdatx: .ds. 1 1 rotdaty: .ds. 1 1 rotdatz: . ds. 1 1 rlnumline: .ds .w 1 rlnummark: .ds. w 1 rlnumfla: . ds. w 1 plusagle: . ds. w 1 rldatx: .ds. w 1540 rldaty: . ds. w 1540 rldatz: .ds. w 1540 rllin: .ds. 1 3200 .data * for every line 4-Bytes *********************************************************************** * These are the coordinates of the definition line which * * generates the rotation body through rotation about * * the Y-axis. By changing coordinates the body to be * * created can be changed. Of course, the number of points in * * rlnum pt must be adapted to the new situation. By changing * * rlnumro the current body can be changed as well. * * Storage reserved here is enough for a maximum 120 rotations * * of 12 points. This means that for a user-defined * * rotation line, the product of the number of points and * * number of desired rotations plus one, cannot be greater * * than 1500. * *********************************************************************** rlxdat: .dc.w 0,40,50,50,20,30,20,30,70,80,80,0 rlydat: -dc.w 100,100,80,60,40,30,30,-70,-80,-90,-100,-100 rlzdat: .dc.w 0,0,0,0,0,0,0,0,0,0,0,0 rlnumpt: .dc.w rlnumro: .dc.w 12 8 * Number of rotations for creation 184 Abacus Software ST 3D Graphics * Definition of the house * * * .data housdatx: .dc.w .dc.w housdaty: .dc.w .dc.w .dc.w housdatz: .dc.w .dc.w .dc.w houslin: .dc.w .dc.w .dc.w .dc.w hnummark: .dc.w hnumline: .dc.w hxangle: .dc.w hyangle: .dc.w hzangle: .dc.w xwplus: .dc.w ywplus: .dc.w zwplus: .dc.w picturex: .dc picturey: . dc -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 30,30,30,30,30,30,30,30,30,30,30,30 30,30,-30,-30,30,30,-30,-30,70,70,-30,0,0,-30 20 , 20 , 0 , 0 , 20 , 20 , 0,0 -10,-10,-30,-30 60,60,60,60,-60,-60,-60,-60,60,-60,60,60,60,60 40,10,10,40,-10,-40,-40,-10 0 ,- 20 ,- 20,0 1,2,2, 3,3, 4,4,1,2,5,5,8, 8,3, 8, 7,7,6, 6,5,6,1,7, 4 9,10,1,9,9,2,5,10,6,10,11,12,12,13,13,14 15,16,16,17,17,18,18,15,19,20,20,21,21,22,22,19 23,24,24,25,25,26,26,23 26 * Number of corners in the house 32 * Number of lines in the house 0 * Rotation angle of house about X-axis 0 * .in « Y-axis 0 * " " " Z-axis 0 * Angle increment around X-axis 0 * Angle increment around Y-axis 0 * Angle increment around Z-axis 0 * Definition of zero point of the screen 0 * provided with values from subroutine getreso 185 ST 3D Graphics Abacus Software rotdpx: • dc.w 0 rotdpy: .dc.w 0 rotdpz: .dc.w 0 rlzl: .dc.w 0 normz: .dc.w 1500 .bss plusrot: .ds. 1 1 first: . ds. w 1 second: .ds. w 1 deltal: .ds . w 1 .data flag: .dc.b 1 .even .bss diffz: .ds. w 1 dx: .ds .w 1 dy: .ds. w 1 dz : .ds. w 1 worldx: . ds. w 1600 * World coordinate array worldy: .ds. w 1600 worldz: . ds. w 1600 viewx: .ds. w 1600 * View coordinate array viewy: .ds. w 1600 viewz: .ds. w 1600 screenx: ,ds. w 1600 * Screen coordinate array screeny: . ds. w 1600 186 Abacus Software ST 3D Graphics wlinxy: .ds. 1 3200 * Line array • data prox: . dc. w 0 * Coordinates for projection- proy: . dc. w 0 * center here on the positive proz: .dc. w 1500 * Z-axis .data of fx: .dc.w 0 ★ Transformation for rotation of fy: . dc. w 0 * to point [offx,offy,offz] of f z : .dc.w 0 xoffs: .dc.w 0 * Inverse transformation for point yoffs: .dc.w 0 ★ [xoff,yoffs, zoffs] zoffs: .dc.w 0 .bss loopc: .ds. 1 .end 1 187 ST 3D Graphics Abacus Software 4.2.1 New subroutines in this program: r 1 set: Supplies the rotation body creation routine with the parameters of the specific rotation body, i.e. with the address of its definition line, with the number of the points forming this line and the desired number of rotations. makerotl: Creates the rotation body rotl in the array rldatx, rldaty, rldatz, and the lines (rllin) and passes the total number of points and lines created. rotstart: Creates the points of the rotation body and is called by makerotl as is: rot 1 in: Creates the lines of the rotation body. wrld2set: Passes the parameters of the world system and the rotation body to the link file variables. The variables for storing of the rotation angle hxangle remain the same, nothing in inp_chan needs to be changed. In contrast to the first program where the house was already explicitly provided, the object to be represented must first be created. This is the task of the subroutine makerotl, which generates the rotation body in the array rldatx, rldaty, rldatz. This array corresponds to the house array housdatx, housdaty, housdatz. The rotation body is transferred to the world system and its position parameters in the main loop are modified in a loop. You should experiment freely with this program and change the definition line for the rotation body and the number of rotations. The only limitation is in the maximum number of points and lines where the total number of lines rlnumline is calculated as follows: rlnummark: Total number of comers in the rotation body rlnumline: Total number of lines in the rotation body r lnumpt : Number of points in the definition line 188 Abacus Software ST 3D Graphics r lnumro: Number of desired rotations of the definition line rlnumline:= ((rlnumpt - 1) * (rlnumro) + (rlnumpt * rlnumro)) rlnummark:= (rlnumpt * (rlnumro + 1)) The number of points can not exceed 1600 and the number of lines cannot be greater than 3200. The expression (rlnumro+1) results from the programming trick, of rotating the definition line one time more than necessary. The definition line, which is the first line in the array, is created a second time at the end of the array. This simplifies the construction of the line array. And now you can try the various rotation lines such as the following: Definition of a Ball: * Definition line and parameter of the ball * * from Fig. 4.2.1 ************ rlxdat: .dc.w rlydat: .dc.w rlzdat: .dc.w rlnumpt: .dc.w rlnumro: .dc.w for creation 0,40,70,90,100,90,70,40,0 100,90,70,40,0,-40,-70,-90,-100 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0,0 9 60 * Number of rotations You need only exchange the corresponding lines in the listing for these. The operation parameters of the program are the same as in house 1: cursor left and right: Change the Y-rotation angle increment cursor up and down: Change the X-rotation angle increment undo and help: Change the Z-rotation angle increment 189 ST 3D Graphics Abacus Software + and - on the keypad: Move the projection plane on the Z-axis (increase or decrease the size of object). * and / on the keypad: Move the rotation reference point on the Z-axis Shift 'D': Hardcopy on the printer 190 Abacus Software ST 3D Graphics 4.3 Hidden line algorithm for convex bodies If you are familiar with real time 3-D graphics on other computers, you were probably surprised by the speed of the display of the wire frame drawings on the Atari ST. On the other hand some game freaks may remark that "I’ve seen the fastest 3-D games on my 8-bit C-64 and these wire models just don’t compare.” For game programming, the main emphasis is on the desired effect. Therefore the active figures for these 3- D-Games are mostly space ships and landscapes which are pre-calculated and their point coordinates are already stored in the computer. For the display which follows on the screen, the object is simply drawn, which naturally can be done quickly, even with 8-bit computers. A disadvantage of this method is the enormous storage requirement, since every possible position of the object must be available in memory, meaning that this procedure cannot be used with complex bodies. In this case only the rotation matrices for the rotation around three axes are calculated ahead of time and stored in a table. Even with this method the limits of the storage are reached quickly. An extreme example: If you want to calculate the rotation matrices of all possible values for subsequent rotation about three axes, with an angle increment of one degree previously calculated, the result will be more than 46 million possibilities (variations of three rotations around 360 possible angles). If this method is used, the degree of freedom of the objects must be limited to one or two possible axes, and/or the gradations of the angle values must be raised so that the table is calculated, for example, only in ten degree steps, or only rotations from zero to to ninety degrees are permitted. Another common method consists of defining the objects as picture shapes, quasi-sprites, in various positions and to switch back and forth between the various shapes and to move the whole shape over the display. Of course the last procedure is the fastest since nothing has to be calculated and the only operation is moving data into the screen memory. Now back to the Atari ST, which, because of its enormous computing power, can not only calculate the wire frame drawing in real time, but as you will see also offers the ability to display simple convex bodies in real time without the hidden lines. The method used corresponds to the surface method used in chapter 2.7. To use this method you must specify every surface of the object precisely. For the example of our house, we need two new variables. First the number of surfaces of the house (hnumpla=13), and second the storage space for the description of these surfaces (houspla). Every surface is described by the number of 191 ST 3D Graphics Abacus Software lines pertaining to it, followed by the lines themselves. The description: 4,1,2,2,3,3,4,4,1 would mean: Four lines belong to this surface and appear as follows: Line #. connects Point # with Point # 1 1 2 2 2 3 3 3 4 4 4 1 To return to the example of our house, it will be necessary to describe all of the surfaces of this house in the same manner. For this reason we draw the various views of the house and number the surfaces in any desired sequence as in Figures 4.3.1 to 4.3.6. In these illustrations the desired result is already achieved, i.e. the hidden lines are already removed to prevent confusion. Figure 4.3.1 - 4.3.6: Hardcopy of House Views 192 Abacus Software ST 3D Graphics ST 3D Graphics Abacus Software point. To make it possible for the algorithm to recognize the hidden surfaces, the sequence of the line points (the direction of the individual lines) is not arbitrary but must be done in the clockwise direction. This is the procedure: 1. Number the surfaces. 2. Create a surface array containing the number of lines (counted clockwise) of each surface as well as the lines of each surface, as viewed from the outside. 3. When all surfaces have been taken care of the number of surfaces are stored in a variable (numpla). Here is the surface list for the thirteen surfaces of the house from Figure 4.3.1. You can get the point indices from Figure 4.1.3. Surface # Number Lines Lines from Point # to Point # 1 4 1, 2 2, 3 3/ 4 4, 1 2 4 2, 5 5, 8 8, 3 3, 2 3 4 5, 6 6, 7 7, 8 8, 5 4 4 7, 6 6, 1 1/ 4 4, 7 5 4 4, 3 3, 8 8, 7 7, 4 6 4 2, 9 9, 10 10, 5 5, 2 7 4 10, 9 9, 1 1, 6 6,10 8 3 1/ 9 9, 2 2, 1 9 3 5, 10 10, 6 6, 5 10 4 11, 12 12,13 13, 14 14,11 11 4 15,16 16, 17 17,18 18,15 12 4 9,20 20,21 21,22 22,19 13 4 23,24 24,25 25,26 26,23 Number of surfaces: 13 With this method of surface definition you can describe up to 32,000 lines which can be the connecting lines for 16,000 different points, though only if you have enough memory, of course. The actual main program hidel. s corresponds to the first main program house 1. s. Two subroutines have been added: hideit: and surfdraw: and two 196 Abacus Software ST 3D Graphics other changes were made in the main loop. The subroutine hide it determines which surfaces are visible from the projection center with the help of the information in the surface arrary (wplane). The information on the visible surfaces, which correspond to the normal surfaces in the structure, first the number of lines followed by individual lines, is entered into a second array (vplane) and the total number of visible surfaces is stored in the surface counter surfcnt. All visible surfaces are subsequently drawn on the display by the subroutine surf dr aw: whereby many lines are drawn twice since the subroutine surf dr aw: takes the lines to be drawn directly from the surface array (vplane). Figure 4.3.1 and the connecting lines of points 2 and 3 show a concrete illustration. This connecting line belongs to the visible surface 1 and the visible surface 2. Naturally all the lines in the surface array (vplane) could be sorted before drawing and double lines removed. My experience shows that the time saved in drawing is lost in the additional sorting and testing, at least for less complicated bodies. Furthermore, the surface information is lost by the separation of the lines, which is needed in the following program sections. Again to run this program you must first compile and link it to grlinkl.s using the batch.ttp file and entering: aslink grlinkl hidel Desk File View Options »' - . I" " 0 :\ 4 BASIC1C0 S S BATCH TTP BUS i iifl bib Inlli BAT COLOR BAK COLOR PRG COLOR S COLOR1 BAK COLOR1 0 COLOR1 S m w-1 r- im □ -is □ E: \ F:\3DW0RK.DIRS ! B butes used X ASStLPRO HIDEl S X FILE-PRO H0USE1 PRG X FORTOT H0USE1 S X GEHDRAM MAIH1 PRG — laTMt ? OPEN APPLICATION Kane: BATCH Parameters: . aslink grlinkl hldelj- • TTP Cancel I rIKICO pro PIN1C0 S HU1 PRO NU1 ILTI1 mil IINT1 IINT1 S PRO S PRG S R0TRTE1 PRG ROTATEl S W a* a" a*. was Ol n -■! '."il'M ;irna nw rrtm flCCBESXMi 197 ST 3D Graphics Abacus Software * hidel.s 19.1.86 Version 3.0 * * House with hidden-line algorithm * * * .globl main,xoffs,yoffs,zoffs,offx,offy,offz -globl viewx,viewy,viewz .globl wlinxy,mouse off,setrotdp,inp chan,pointrot .text main: jsr apinit * Announce program jsr grafhand * Get screen handler jsr openwork * Display jsr mouse off * Turn off mouse jsr getreso * what resolution ? jsr setcocli * Prepare clip window move.1 #houspla. worldpla * Address of surface array jsr makewrld * Create world system jsr wrldset * Pass world parameters jsr setrotdp * initialize observer ref. point jsr clwork jsr pagedown * Display logical page jsr clwork jsr inp_chan * Input and change parameters mainlopl: jsr pointrot * rotate about observer ref. point jsr pers * Perspective transformation jsr hideit jsr surfdraw jsr pageup * Display physical page jsr inp chan * Input new parameters jsr clwork jsr pointrot * Rotate around rotation ref. point jsr pers * Transform new points jsr hideit jsr surfdraw 198 Abacus Software ST 3D Graphics jsr jsr jsr jmp pagedown inp_chan clwork mainlopl * Display this logical page * Input and change parameters * erase physical page * to main loop mainend: move.1 physbase,logbase jsr rt s pageup * switch to normal display page * back to link file, and end * Input and change parameters such as angle increments and * Z-coordinate of the projection plane inp_chan: jsr inkey cmp.b #'D',dO bne inpwait jsr scrdmp inpwait: swap dO cmp.b #$4d,dO bne inpl addq. w #1,ywplus bra inpendl inpl: cmp.b #$4b,dO bne inp2 subq.w #1,ywplus bra inpendl inp2: cmp.b #$50,dO bne inp3 addq.w #1,xwplus bra inpendl inp3: cmp.b #$48,dO bne inp3a subq.w #1,xwplus bra inpendl * Sense keyboard, keyboard code in * Make harcopy * Test DO for * Cursor-right * if yes, then add one to * Y-angle increment and continue * Cursor-left, if yes * then subtract one from * Y-angle increment * Cursor-down, if yes * then add one to X-angle increment * Cursor-up * subtract one 199 ST 3D Graphics Abacus Software inp3a: cmp.b #$61,dO bne inp3b subq * w #1,zwplus bra inpendl inp3b: cmp.b #$62,dO bne inp4 addq.w #1,zwplus bra inpendl inp4: cmp.b #$4e,dO bne inp5 sub. w #25,dist bra inpendl inp5: cmp.b #$4a,dO bne inp6 add.w #25,dist bra inpendl inp6: cmp.b #$66,dO bne inp7 sub. w #15,rotdpz bra inpendl inp7 : cmp.b #$65,dO bne inplO add.w #15,rotdpz bra inpendl inplO: cmp.b #$44,dO bne inpendl addq.1 #4,a7 bra mainend inpendl: move.w hyangle,dl add. w ywplus,dl cmp. w #360,dl bge inpend2 cmp. w #-360,dl ble inpend3 bra inpend4 * Undo key * Help key * + key on keypad * if yes then subtract 25 from * location of projection plane * (Z-coordinate) * - key on keypad * * if yes then add 25 * * key on keypad * if yes, subtract 15 from the * rotation point Z-coordinate * Make change * / key of keypad * Add 15 * F10 pressed ? * if yes, jump to * program end * Rotation angle about Y-axis * add increment * if larger than 360, subtract 360 * if smaller than 360 * add 360 200 Abacus Software ST 3D Graphics inpend2: sub. w #360,dl bra inpend4 inpend3: add.w #360,dl inpend4: move.w dl,hyangle move.w hxangle,dl add.w xwplus,dl cmp. w #360,dl bge inpend5 cmp. w #-360,dl ble inpend6 bra inpend7 inpend5: sub. w #360,dl bra inpend7 inpend6: add.w #360,dl inpend?: move.w dl,hxangle move.w hzangle,dl add.w zwplus,dl cmp. w #360,dl bge inpend8 cmp. w #-360,dl ble inpend9 bra inpendlO inpend8: sub.w #360,dl bra inpendlO inpend9: add.w #360,dl inpendlO : move.w dl,hzangle rts * Treat * rotation angle about X-axis * in the same manner * Initialize the rotation reference point to [0,0,0] and the * rotation angle also to 0,0,0 ************************************************** ******************** setrotdp: move.w move. w move.w move.w #0, dl dl,rotdpx dl,rotdpy dl,rotdpz * set the start rotation- * datum point 201 ST 3D Graphics Abacus Software move.w #0,hyangle * Start rotation angle move.w #0,hzangle move.w #0,hxangle rts ************************************************************************ * Rotate the total world system around one point, the rotation * * reference point * ************************************************************************ pointrot: move.w hxangle,xangle * rotate the world around the move.w hyangle,yangle move.w hzangle,zangle move.w rotdpx,dO * rotation reference point move.w rotdpy,dl move.w rotdpz,d2 move.w dO,xoffs * add for inverse transformation move.w dl,yoffs move. w d2,zoffs neg. w dO neg. w dl neg. w d2 move.w dO,offx * subtract for transformation move.w dl,offy move.w d2,offz jsr matinit * Matrix initialization jsr zrotate * first rotate about Z-axis jsr yrotate * rotate 'matrix' about Y-axis jsr xrotate * then about X-axis jsr rotate * Multiply points with matrix rts *********************************************************************** * Generate world system from object data. All points, lines, * * and surfaces are transferred to the world system * *********************************************************************** makewrld: move.1 move. 1 move.1 move.1 move.1 #housdatx,al ihousdaty,a2 #housdatz,a3 twrldx ,&A #wrldy,a5 * Generate world system by 202 Abacus Software ST 3D Graphics move.l #wrldz,a6 move.w hnummark,dO ext.l dO subq.l #l,dO makewll: move.w (al)+, (a4) + move.w (a2)+, (a5)+ move.w {a3)+,(a6) + dbra dO,makewll move.w hnumline,dO ext.l dO subq.1 #1/dO move.l #houslin,al move.l #wlinxy,a2 makewl2: move.l (al)+,(a2)+ dbra d0,makewl2 move.l worldpla,aO move.l #wplane,al move.w hnumsurf,dO ext.l dO subq.l #l,dO makewl3: move.w (aO)+,dl move.w dl,(al)+ ext.l dl subq.l #l,dl makewl4: move.l (a0)+, (al) + dbra dl,makewl4 dbra d0,makewl3 rts * Copying point coordinates * to world system * Number of house lines * Copy all lines into * world system * Number of surfaces on house * Copy all surface * definitions into the * world system * Copy every line of this * surface into the world array * until all surfaces are processed * Passing the world parameters to the link file variables wrldset: move.l #wrldx,datx * Pass variables for move.l #wrldy,daty * the rotation routine move.l #wrldz,datz move.l ^viewx,pointx 203 ST 3D Graphics Abacus Software move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,linxy move.w picturex,xO move.w picturey,yO move.w proz,zobs move.w rlz1,dist move.1 tscreenx,xplot move.1 tscreeny,yplot move. w hnumline,numline move.w hnummark,nummark move.w rts hnumsurf,numsurf * remove all characters from the keyboard buffer * clearbuf: clearnd: move.w #$b,-(a7) trap #1 addq.1 #2, a? tst. w dO beq clearnd move. w #1,-(a7) trap #1 addq.1 #2, a7 bra clearbuf rts ********************************************************************* * Sense display resolution and set coordinate origin of screen * * to screen center * ********************************************************************* move.w #4,-<a7) * Sense screen resolution trap #14 addq.1 #2,a7 cmp. w #2, dO bne getrl move.w #320,picturex * Monochrome monitor 204 Abacus Software ST 3D Graphics move.w #200,picturey bra getrend getrl: cmp.w #l,d0 bne getr2 move.w #320,picturex * medium resolution (640*200) move.w #100,picturey bra getrend getr2: move.w #160,picturex * low resolution (320*200) move.w #100,picturey getrend: rts *********************************************************************** * Hardcopy routine, called by inp_chan *********************************************************************** scrdmp: move.w trap addq.1 jsr rts #20,-(a7) #14 #2.a7 clearbuf ********************************************************************* * Sets the limits of the display window for the Cohen-Sutherland * clip algorithm built into the draw-line algorithm. * The limits can be freely selected by the user, which makes the * draw-line algorithm very flexible. **************** move . w #0,clipxule move.w #0,clipyule move.w picturex,dl lsl. w #l,dl * times two subq.w #l,dl * minus one equal move.w dl,clipxlri * 639 for monochrome move.w picturey,dl lsl. w #1, dl * times two minus one equal subq.w #1, dl * 399 for monochrome move.w dl,clipylri rts 205 ST 3D Graphics Abacus Software *********************************************************************** * Recognition of hidden surfaces and entry of these into the * * vplane array, the surface information is in the surface array * * wplane, as well as in view system, viewx, viewy, viewz, * * also the total number of surfaces must be passed in numsurf * *********************************************************************** hideit: visible: move.w numsurf,dO i ' Number of surfaces as counter ext .1 dO subq.1 #1, dO move.1 #viewx,al * Store point coordinates here move.1 #viewy,a2 move.1 #viewz,a3 move.1 Iwplane,aO * Information for every surface move.1 #vplane,a5 * here. move.w #0, surfcount * counts the known visible surfaces. move.w (aO),dl * start with first surface, number ext. 1 dl * of points of this surface in Dl. move.w 2(aO),d2 * Offset of first point of this surf. move.w 4(aO),d3 * Offset of second point move.w 8(aO),d4 * Offset of third point subq.w #1, d2 * for access to point arrays subtract subq.w #1, d3 * one from current point offset subq.w #1, d4 * multiply by two lsl. w #l,d2 lsl. w #1, d3 lsl. w #l,d4 * and finally access current point move.w (al,d3.w),d6 * coordinates cmp. w (al, d4 . w) , d6 * comparison recognizes two points bne doitl * with same coordinates which can move.w (a2,d3.w),d6 * result during construction of cmp. w (a2,d4.w),d6 * rotation bodies. During recognition bne doitl * of two points in which all point move.w <a3,d4.w),d6 * coordinates match (x,y,z) the cmp.w (a3, d3 . w) , d6 * program selects a third point for bne doitl * determination of the two vectors move.w 12(aO),d4 subq.w #l,d4 lsl. w #1, d4 206 Abacus Software ST 3D Graphics doit1: move.w (al,d3.w),d5 move.w d5, kx sub. w (al,d2.w),d5 move.w d5,px move.w (a2,d3.w),d5 move.w d5,ky sub. w (a2,d2.w),d5 move.w d5,py move.w (a3, d3 . w) , d5 move.w d5, kz sub. w (a3,d2.w), d5 move.w d5, pz move.w (al, d4.w),d5 sub. w (al, d2.w),d5 move.w (a2, d4.w),d6 sub. w (a2,d2.w) ,d6 move.w (a3,d4.w),d7 sub.w (a3,d2.w),d7 move.w d5,dl move. w d6,d2 move.w d7,d3 nulls py,d3 mu Is pz,d2 sub.w d2,d3 move.w d3, rx mu Is pz,dl mu Is px,d7 sub. w d7,dl move.w dl, ry mu Is px, d6 mu Is py,d5 sub. w d5,d6 move.w d6, rz move.w prox,dl sub.w kx,dl move. w proy,d2 sub.w ky,d2 move.w proz,d3 * Here the two vectors, which lie * in the surface plane, are * determined by subtracting the * coordinates of two points * from this surface. * The direction coordinates of the * vectors are stored in the * variables qx,qy,qz and px,py,pz * Calculate vector Q * qx * qy * qz * Calculate the cross product * of the vertical vector for the * current surface. * The direction coordinates of the * vertical vector are stored * zobsorarily in rx,ry,rz * The projection center * is used as the comparison * point for the visibility * of a surface. * One can also use the 207 ST 3D Graphics Abacus Software sub. w kz, d3 * mu Is rx, dl * mu Is ry, d2 * mu Is rz, d3 * add. 1 dl, d2 * add. 1 d2,d3 * bmi dosight * * If the scalar product is negative. move.w (aO),dl * ext. 1 dl lsl.l #2, dl * addq.1 #2, dl * add. 1 dl, aO * sightl: dbra dO,visible * bra hideend * dosight: move. w (aO),dl * ext. 1 dl * lsl.l #1, dl * sight3: move. w (aO)+,(a5)+ * dbra dl,sight3 * addq. w #1, surfcount * bra sightl * hideend: rts observation ref. point as the comparison point. Now comes the comparison of vector R with the vector from a point on the surface to the projection center for creating the scalar product of the two vectors. the surface is visible Number of lines of the surface Number of lines times 4 = space for lines plus 2 bytes for the number of lines added to surface array, for access to next surface. When all surfaces completed then end. Number of lines for this surface, gives the number of words to be transmitted when multiplied by 2. pass the number of lines and the the individual lines the number of surfaces plus one and process the next * Draw visible surfaces passed in vplane * surfdraw: move.l xplot,a4 move.l yplot,a5 * Draws a number of surfaces (passed * in surfcount) whose description move.1 move.w #vplane,a6 surfcount,dO * is in the array at address * vplane, and was entered by routine 208 Abacus Software ST 3D Graphics ext. 1 dO * hideit subq.1 #1, dO * if no surface is entered in the bmi surfend * array, then end. surflopl: move. w (a6)+,dl * Number of lines in this surface as ext. 1 dl * counter of lines to be drawn. subq. 1 #1, dl surflop2: move. 1 (a6) +,d5 * First line of this surface subq.w #1, d5 * Access screen array which contains lsl. w #1, d5 * screen coordinates of the points. move . w 0(a4,d5.w),d2 move. w 0(a5,d5.w),d3 * extract points from routine and swap d5 * pass. subq.w #1, d5 lsl .w #1, d5 move.w 0 (a4,d5 .w) , a2 * second point of line move.w 0(a5,d5.w),a3 jsr drawl * Draw line until all lines of this dbra dl,surflop2 * surface have been drawn and repeat dbra dO,surflopl * until all surfaces are drawn. surfend: rts * Return. * Here begins the variable area of the program module * * * * * * Definition of the house * * * .data housdatx: .dc.w • do. w -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 30,30,30,30,30,30,30,30,30,30,30,30 209 ST 3D Graphics Abacus Software housdaty: .dc, , w 30, 30,-30,-30,30,30, -30, -30, 70, 70,- 30, o to 1 o o . dc, . w 20, 20, 0,0,20,20,0,0 . dc. . w -10 ,-10,-30,-30 housdatz: • dc. , w 60, 60,60,60,-60,-60, -60, -60, 60, -60, 60, 60, 60, 60 .dc, . w 40, 10,10,40,-10,-40, -40, -10 . dc. . w o,- 20,-20,0 houslin: . dc. . w 1, 2 ,2,3,3,4,4,1,2,5, 5,8, 8,3, 8, 7 ,7, 6 , 6, 5, 6, 1,7,4 . dc. , w 9, 10,1, 9, 9,2,5,10, 6, 10,11,12 , 12 ,13, 13, 14 . dc, . w 15, 16,16,17,17,18,18 ,15, 19,20,20,21 ,21 ,22,22,19 . dc. . w 23, 24,24,25,25,26,26 ,23 * here are the definitions of the surfaces belonging to the house * houspla: .dc.w .dc. w .dc.w .dc.w .dc.w .dc.w • dc.w hnummark: .dc.w hnumline: .dc.w hnumsurf: .dc.w hxangle: .dc.w hyangle: .dc.w hzangle: .dc.w xwplus: .dc.w ywplus: .dc.w zwplus: .dc.w picturex: .dc.w picturey: .dc.w 4,1,2,2,3,3,4,4,1,4,2,5,5,8,8,3,3,2 4,5,6, 6, 7, 7,8, 8,5, 4,7,6, 6,1,1, 4,4,7 4,4,3,3,8, 8,7, 7,4,4,2,9, 9,10, 10,5, 5,2 4,10,9,9,1,1,6,6,10,3,1,9,9,2,2,1 3,5,10,10,6,6,5,4,11,12,12,13,13,14,14,11 4,15, 16, 16, 17, 17, 18, 18, 15,4, 19, 20,20, 21,21, 22, 22, 19 4,23,24,24,25,25/26,26,23 26 * Number of corner points of the house 32 * Number of lines of the house 13 * Number of surfaces of the house 0 * Rotation angle of house about X-axis 0 * " " Y-axis 0 * " " Z-axis 0 * Angle increment about X-axis 0 * Angle increment about Y-axis 0 * Angle increment about Z-axis 0 * Definition of zero point of display 0 * entered by getreso 210 Abacus Software ST 3D Graphics rotdpx: • dc. w 0 rotdpy: . dc. w 0 rotdpz: .dc. w 0 rlzl: . dc. w 0 normz: .dc. w 1500 .bss plusrot: . ds. 1 1 first: .ds.w 1 second: .ds. w 1 deltal: .ds.w 1 worldpla: .ds. 1 1 * Address of surface array .data plag: .dc .b 1 .even .bss diffz: .ds.w 1 dx: .ds.w 1 dy: .ds.w 1 dz: .ds.w 1 wrldx: .ds.w 1600 * World coordinate array wrldy: .ds.w 1600 wrldz : .ds.w 1600 viewx: .ds.w 1600 * View coordinate array viewy: .ds.w 1600 viewz: .ds.w 1600 screenx: .ds.w 1600 * Display coordinate array screeny: .ds.w 1600 211 ST 3D Graphics Abacus Software wlinxy: • ds. 1 3200 * Line array wplane: .ds. 1 6600 * Surface array vplane: .ds. 1 6600 * Surface array of visible surfaces surfcount: • ds. w 1 numsurf: • ds. w 1 zcount: . ds. 1 1 * Sum of all Z-coordinates zpla: . ds. w 1 * Individual Z-coordinates of surface sx: . ds. w 1 sy: .ds .w 1 sz: .ds .w 1 px: .ds. w 1 py: • ds. w 1 pz: • ds. w 1 rx: .ds. w 1 ry: .ds. w 1 rz: . ds. w 1 qx: .ds .w 1 qy: -ds. w 1 qz: .ds. w 1 kx: -ds. w 1 ky: .ds. w 1 kz: .ds. w 1 . data prox: .dc.w 0 * Coordinates of the projection center proy: . dc. w 0 * on the positive Z-axis proz: .dc.w 1500 212 Abacus Software ST 3D Graphics .data of fx: . dc. w 0 * Transformation during rotation of fy: .dc. w 0 * to point [offx,offy,offz] of f z: . dc. w 0 xoffs: .dc. w 0 * Inverse transformation to point yoffs: . dc. w 0 ★ [xoff,yoffs, zoffs] zoffs: .dc. w 0 .bss loopc .ds. 1 . end 1 213 ST 3D Graphics Abacus Software 4.3.1 Explanation of the newly-added subroutines hideit: In contrast to the explanation in the mathematical part, the view system used by the program is a right system; this saves the multiplication of the Z-values by -1. The subroutine hideit forms two vectors within the surface from the first three points of every surface. These are the vectors from point one to point two as well as the vector from point one to point three. These two vectors correspond to the vectors P[px,py,pz] and Q[qx,qy,qz] from chapter 2.7. Furthermore, a third vector R[rx,ry,rz] is generated through the formation of the cross product of the vectors P and Q. According to the definition, the cross product is perpendicular to the vectors P and Q and, in this sequence forms a right-hand system with them [p, q, r]. Finally, a vector is created from a point on the surface to the projection center (S[sx,sy,sz]), and its direction is compared with the direction of the vector R by creation of the scalar products of the vectors S and R. All the surfaces which are in front of the projection center are visible. Scalar product= sx*rx+sy*ry+sz*rz = !s|*|r|*cos(Alpha) Alpha is the angle suspended between the vectors R and S. If the result of the scalar product is negative, this means an angle larger than 90 degrees and smaller than 270 degrees between the two vectors, which point in different directions (See also Figure 2.7.1), and so this surface is visible, according to the surface definition (clockwise direction) and right system used. surf dr aw: Here the visible surfaces are displayed by drawing the lines of the array vplane. The whole job was done already by hideit. The operation parameters of the program are the same as in house 1. s, The rotation point on the Z-axis can be moved with the * and / keys on the keypad, the projection plane can be moved with the - and + keys on the keypad, and the angle increments of the rotation angle around the X 214 Abacus Software ST 3D Graphics and Y-axis can be changed with the cursor keys and the Help and Undo keys. Of course you can also change all the parameters within the program (projection center, rotation reference point to X and Y-axis, etc.). 4.3.1.1 Errors with non-con vex bodies If the rotation creation routine is added to the main program and the chess figure is created with hideit: and pladraw without hidden lines: you can see the problem. With concave bodies such as this chess figure there is the possibility that one of the surfaces recognized by the hideit: algorithm as visible can be hidden by another visible surface during viewing. In this case the hideit: algorithm fails and the problem must be solved with another algorithm. 215 ST 3D Graphics Abacus Software 4.4 The painter algorithm Recall the problem we‘re trying to solve: Surfaces which are seen from an observation point have their surface normal vector pointed in another direction from a vector from any point on the surface to the projection center, are hidden by other surfaces which according to this criterium are also visible. If you start from the observation point (projection center) on the positive Z-axis, the middle Z-coordinate of a surface is a possible description of it and its position in the world system. The middle Z- coordinate is obtained by defining the arithmetic center of the comer point coordinates belonging to the surface, i.e. summation of all surface comer point Z-coordinates and division by the number of comer points belonging to the surface. The relationship can be made clear with the simple example with three different surfaces in Figure 4.4.1. 216 Abacus Software ST 3D Graphics Viewing the defined world system from one point on the positive Z axis, we can say: the surface with the largest middle Z-coordinate is visible in its entire size and is not hidden by any other surface. Note that all observed surfaces are on the negative Z-axis (-200 > -400). This completely visible surface covers parts of surfaces with a smaller middle Z coordinate. Surfaces 2 and 3 are covered by surface 1 and surface 3 is again covered by surface 2. The surfaces are sorted by their Z-coordinates and they are drawn starting with the smallest middle z-coordinate, surface 3, and then the surfaces with the larger Z coordinates, and we have found a possible solution to the problem by covering hidden surfaces with other surfaces. You must consider that it is not enough just to draw every surface. The individual surfaces must be filled with "color" or a pattern so that the surfaces drawn first are really covered. Figure 4.4.2 shows one possible result + Y Figure 4.4.2 If we think about our rotation body from chapter 4.2, this means first of all that when the rotation body is created its surfaces must also be 217 ST 3D Graphics Abacus Software created, second a middle Z coordinate must be calculated and stored some place for every surface. Another problem is sorting the surfaces. If one wanted to sort every defined form with its lines, it would require an enormous movement of data in storage. To avoid this, a new storage area is created in which the Z-coordinates together with the beginning address of the surface it pertains to are stored. The individual surfaces are stored in a simple linear list. The beginning address of every surface is the storage address at which the number of lines for this surface is stored. Through storage of this address, it is possible to access every single surface directly, which previously was not possible because of the number of lines belonging to each surface. To better handle the two pieces of information, (Z-coordinates of the surface and address of the surface) we select a long word as data size for both, i.e. in the newly constructed array (surfaddr) there are four successive bytes for the Z-coordinate and four bytes for the address of the surface. Each description of a surface "occupies" eight bytes of storage space. This array contains the visible surfaces represented by their middle Z-coordinates and their beginning addresses in the new addition to the subroutine hide it: (sight 2). In this special case of the rotation body whose surfaces all consist of four lines, the division by the line number (4) for calculation of the middle Z-coordinate can be performed by shifting right by two bit positions. If you want to include surfaces with more or less than four lines in the paint routine, you must alter the hide it -routine and divide by the number of surfaces. After the adaptation of the subroutine hideit: all visible surfaces are in the two arrays, in vplane: and in surfaddr:. The number of surfaces, like in the first version of hideit:, goes in the variable placount:. Fortunately, we do not have to write the shading function since the operating system offers a function for filling display areas with a shading pattern (Filled Area). This function fills a polygon whose points are passed in the ptsin array, with one of a total of 36 different predefined, and one user-defined shading pattern. Before calling this function with the opcode 9, we set up the different shading parameters which is done using the subroutines filmode, filform, filcolor, filstyle and f ilindex which are contained in the link file (grlink 1). The shading routine is called by the subroutine paint it, which first sorts all surfaces contained in surfaddr: according to ascending Z- coordinates. Next you must pass the individual surfaces, i.e. their end point coordinates, to the function "Filled Area". This begins with the surface which has the smallest middle Z-coordinate. The function "Filled 218 Abacus Software ST 3D Graphics Area" can, in connection with the function "Set Clipping Rectangle", Opcode 129, fill surfaces limited to a display window. It is necessary to call the function "Set Clipping Rectangle" when the display window is the total screen area, bordered by the coordinates 0,0 and 639,399 (for BW monitors), if this is not done, "Filled Area" may draw parts of the polygon sticking beyond the display frame on the neighboring display page (wrapping). You could fill all surfaces with the same pattern, which could also be white. You can assign a shading pattern for every surface corresponding to its Z-coordinates. We will limit ourselves to only six of the 36 possible fill patterns. This is done purely for optical reasons since shaded surfaces, and even completely filled color surfaces, can have a negative effect on the picture. You can influence this choice or omit it entirely. Simply set the desired pattern on entry to the subroutine. With a color monitor, a various fill colors can be used instead of a shading pattern. The choice of colors is completely up to you. The visual effect of these three-dimensional graphics can best be appreciated with a high- resolution monitor. Doubling the resolution in both directions increases the quality of the picture four times. If you have a color monitor, you can choose between filling with color or patterns. If you want to try filling with color you must call the function filstyle with the value one in the DO register when entering the paint it routine. The subroutine filcolor: makes it possible to use different colors. Owners of monochrome monitors don’t have to change anything in the program. To run this program call the batch file batch.ttp then enter: as link grlinkl paintl 253882 bytes used 1 Si PRINTERS Si TUTORIAL C FRY COM TTP NLlfl PRG OUTPUT PRG SPLIT TTP STRNDARD PRT TEXTPRO PRG TUTORIAL TXT XTTUTORI TOC 1442236 bytes used in 125 itens 38944 OPEN APPLICATION Mane: BATCH .TTP Paraneters: aslink grlinkl paintlj—_ 1 F:\3DM0RK.DIR\ 333956 butes used in H0USE1 PRG H0USE1 S HAIN1 PRG HRIN1 S MAIK1C0 PR6 MAIN1C0 S HENU1 PRG MENU1 S MULTI1 PRG MULTI1 S PAINT1 PRG PAINT1 S R0TATE1 PRG R0TATE1 S ST 3D Graphics Abacus Software Figure 4.4.3 Here is the listing of the fourth main program for the link file grlinkl.s. It is called paintl.s. The operating parameters again correspond to the previous program. 220 Abacus Software ST 3D Graphics ********************************************************************** * paint 1.s 9.2.1986 * * Display a shaded rotation body * * * ★********************************************************************* .text .globl main,xoffs,yoffs, zoffs,offx,offy,offz .globl viewx,viewy,viewz .globl wlinxy,mouse_off,setrotdp,inp_chan,pointrot main: jsr apinit * Announce program jsr grafhand * Get screen handler jsr openwork * open workstation jsr mouseoff * Turn off mouse jsr getreso * Display resolution ? jsr setcocli * Set clip window jsr makerotl * Create rotation body jsr makewrld * Create world system jsr wrld2set ★ Pass world parameters jsr setrotdp ★ initialize observation ref. point jsr clwork jsr pagedown * Display logical page jsr clwork jsr inp chan mainlopl: jsr pointrot * rotate around observ. ref. point jsr pers * Perspective transformation jsr hideit ★ hide hidden surfaces jsr paintit ★ sort and shade jsr pageup * Display physical page jsr inp chan * Input new parameters jsr clwork ★ clear screen page not displayed jsr pointrot * Rotate around rot. ref. point jsr pers ★ Transform new points jsr hideit * hide jsr paint it * sort and shade 221 ST 3D Graphics Abacus Software jsr pagedown * Display this logical page jsr inp chan * Input and change parameters jsr clwork * erase physical page jmp mainlopl * to main loop move. 1 physbase,logbase jsr pageup * Switch to normal screen page rts ★ back to link file and end A********************************************************************** * Creation of rotation body by passing parameters * * and calling rotation routine * it********************************************************************** makerotl: jsr jsr rts rlset * Set parameters of rot. body rotstart * and create rot. body * Input and change parameters with the keyboard * inp__chan: jsr inkey cmp .b #'D',dO bne inpwait jsr scrdmp inpwait: swap dO cmp .b #$4d,dO bne inpl addq. w #1,ywplus bra inpendl inpl: cmp .b #$4b,dO bne inp2 subq. w #1,ywplus bra inpendl * Read keyboard, code in * Make hardcopy * Test DO for * Cursor-right * if yes, add one to * Y-angle increment and continue * Cursor-left, if yes * subtract one from * Y-angle increment 222 Abacus Software ST 3D Graphics inp2 : cmp .b #$50,dO bne inp3 addq. w #1, xwplus bra inpendl inp3: cmp .b #$48,dO bne inp3a subq.w #1,xwplus bra inpendl inp3a: cmp .b #$61,dO bne inp3b subq.w #1,zwplus bra inpendl inp3b: cmp -b #$62,dO bne inp4 addq.w #1,zwplus bra inpendl inp4 : cmp.b #$4e,dO bne inp5 sub. w #25,dist bra inpendl inp5: cmp.b #$4a,dO bne inp6 add. w #25,dist bra inpendl inp6: cmp.b #$66,dO bne inp7 sub. w #15,rotdpz bra inpendl inp7 : cmp.b #$65,dO bne inplO add. w #15,rotdpz bra inpendl * Cursor-down, if yes * add one to X-angle * increment * Cursor-up * subtract one * Undo key * decrease Z-increment * Help key * increase Z-increment * + key on keypad * if yes, subtract 25 from * location of projection * plane (Z-coordinate) * minus key on keypad * * if yes, add 25 * * key on keypad * if yes, subtract 15 from * rotation point Z-coordinate * Make change * / key on keypad * add 15 223 ST 3D Graphics Abacus Software inplO: cmp .b #$44,dO * F10 pressed ? bne inpendl addq. 1 #4,a7 * if yes, then jump to bra ma inend * program end inpendl: move. w hyangle, dl * Rotat.angle about Y-axis add. w ywplus,dl ★ add increment cmp. w #360,dl * if larger than 360, then bge inpend2 * subtract 360 cmp. w #-360,dl * if smaller than 360, then ble inpend3 * add 360 bra inpend4 inpend2: sub. w #360,dl bra inpend4 inpend3: add. w #360,dl inpend4: move. w dl,hyangle move. w hxangle,dl * do the same for add. w xwplus,dl ★ the rotation angle cmp. w #360,dl ★ about X-axis bge inpend5 cmp. w #-360,dl ble inpend6 bra inpend7 inpend5: sub. w #360,dl bra inpend7 inpend6: add. w #360,dl inpend7: move.w dl,hxangle * move.w hzangle,dl add. w zwplus,dl cmp.w #360,dl bge inpend8 cmp. w #-360,dl ble inpend9 bra inpendlO inpend8: sub. w #360,dl bra inpendlO inpend9: add. w #360,dl 224 Abacus Software ST 3D Graphics inpendlO: move.w dl,hzangle rts * Initialize the rotation reference point to [0,0,0] * move.w #0, dl * set the Initial rotation move.w dl,rotdpx * ref. point move. w dl,rotdpy move.w dl,rotdpz move. w #0,hyangle * initial rotation angle move.w #0,hzangle move.w #0,hxangle rts * Rotation around the rotation reference point about all * * three axes * pointrot: move.w hxangle,xan gle * rotate the world around move.w hyangle,yangle move.w hzangle,zangle move.w rotdpx,dO * rotation reference point move.w rotdpy,dl move. w rotdpz,d2 move.w dO,xoffs * add for inverse transform move.w dl,yoffs move. w d2,zoffs neg. w dO neg. w dl neg. w d2 move.w dO,of fx * subtract for transform move.w dl,offy move.w d2,offz jsr matinit * initialize matrix jsr zrotate ■k rotate first about Z-axis jsr yrotate * rotate 'matrix' about Y-axis jsr xrotate * then rotate about X-axis jsr rotate * Multiply points with matrix. rts 225 ST 3D Graphics Abacus Software *********************************************************************** * Create world system by copying the object data into world system * *********************************************************************** makewrld: makewll: makewl2: makew!3: makewl4: move. 1 #rldatx,al * Create world system by move.1 #rldaty,a2 move.1 trldatz,a3 move.1 #wrldx,a4 move. 1 #wrldy,a5 move.1 #wrldz,a6 move.w rlnummark,dO ext. 1 dO subq.1 #l,dO move.w (al)+, <a4)+ * copying point coordinates move.w (a2)+, (a5) + * into the world system move.w (a3)+, <a6)+ dbra dO,makewll move.w rlnumline,dO ext. 1 dO subq.1 #l f dO move.1 #rllin,al move.1 #wlinxy,a2 move.1 (al)+, (a2) + * Copy lines into world dbra d0,makewl2 * system move.1 worldpla,aO move.1 #wplane,al move.w rlnumsurf,dO ext. 1 dO subq.1 #1, dO move.w (aO)+,dl * Copy surfaces into move.w dl, (al) + * world system ext. 1 dl subq.1 #1 / dl move.1 (aO)+,(al)+ * Copy every line of dbra dl,makewl4 * this surface into dbra dO,makewl3 * world array until all rts * surfaces are completed 226 Abacus Software ST 3D Graphics ********************************************************************* * Pass the world parameters to the variables in the * * link files ********************************************************************* wrldset: move.1 #wrldx,datx * move. 1 #wrldy,daty * move.1 #wrldz,datz * move.1 #viewx,pointx move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,linxy move. w picturex,xO move.w picturey,yO move.w proz,zobs move.w rlzl,dist move.1 Iscreenx,xplot move.1 #screeny,yplot move.w hnumline,numline move.w hnummark,nummark move.w rts hnumsurf,numsurf Pass the variables for the rotation routine * Remove all characters from keyboard buffer clearbuf: move. w #$b,-(a7) trap #1 addq. 1 #2, a7 tst. w dO beq clearnd move.w #1,-{a7) trap #1 addq.1 #2, a7 bra clearbuf clearnd: rts 227 ST 3D Graphics Abacus Software ********************************************************************* * Sense display resolution and set coordinate * * origin to screen center * ********************************************************************* getreso: move.w #4,-(a7) * Sense display resolution trap #14 addq.1 #2,a7 cmp. w #2, dO bne getrl move.w #320,picturex ★ Monochrome monitor move.w #200,picturey bra getrend getrl: cmp • w #1, dO bne getr2 move.w #320,picturex * medium resolution (640*200 move.w #100,picturey bra getrend getr2: move.w #160,picturex * low resolution (320*200) move.w #100,picturey getrend: rts * Hardcopy of screen, called by inp_chan * * * scrdmp: move. w trap addq. 1 jsr rts #20,-(a7) #14 #2,a7 clearbuf 228 Abacus Software ST 3D Graphics ********************************************************************* * Sets the limits of the display window for the * Cohen-Sutherland clipping algorithm built into the * draw-line algorithm * The limits can be freely selected by the user which makes ‘ * the draw-line algorithm very flexible. * ********************************************************************* move.w #0, clipxule move . w #0,clipyule move.w picturex,dl lsl. w #l,dl ★ times two subq.w #l,dl ★ minus one equals move.w dl,clipxlri * 639 for monochrom move .w picturey,dl lsl. w #1, dl * times two minus one subq.w #1, dl * equals 399 for monochrome move.w dl,clipylri rts * Pass visible surfaces into vplane array and * into pladress array for subsequent sorting *r * of surfaces hideit: move. w numsurf, dO * Number of surfaces as ext. 1 dO * counter subq.1 #1, dO move.1 #viewx,al * The point move.1 #viewy,a2 * coordinates are stored here move.1 #viewz,a3 move.1 #wplane,aO * Here is the information move.1 fvplane,a5 * for every surface move.w #0,surfcount * Counts the known visible surfaces. move.1 tpladress,a6 * Address of surface storage move.w (aO) ,dl * Start with first surface ext. 1 dl * Number of points on this surface in Dl move. w 2 (aO),d2 * Offset of first point of this surface 229 ST 3D Graphics Abacus Software A doit 1: move.w 4 <a0),d3 * Offset of second point move.w 8 (aO),d4 * Offset of third point subq.w #l,d2 * For access to point array subq.w #1, d3 * subtract one from current subq.w #1, d4 * point offset. lsl. w #1/ d2 * Multiply by two lsl. w #l,d3 lsl -w #1, d4 ★ and access current move. w (al,d3.w),d6 * point coordinates cmp. w (al,d4.w), d6 * Comparison recognizes two point bne doit 1 ★ with the same coordinates ★ created through move.w (a2,d3.w) ,d6 * construction of cmp. w (a2,d4.w) ,d6 ★ rotation bodies. When bne doitl ★ two points are found move.w (a3,d4.w),d6 ★ where all point coordinates (x. cmp. w (a3,d3.w),d6 ★ match, the program selects the bne doitl * third point to find move. w 12(aO),d4 * both vectors subq.w #l,d4 lsl .w #l,d4 move.w (al,d3.w),d5 * the two vectors which move.w d5, kx * lie in the surface plane sub.w (al,d2.w),d5 * are found by subtracting the move.w d5,px * coordinates of two points move.w <a2,d3.w),d5 * in this surface move.w d5, ky * the direction coord, of the sub. w <a2,d2.w),d5 * vectors is stored in move. w d5,py * variables qx,qy,qz and move. w (a3,d3.w),d5 * px,py,pz move. w d5, kz sub. w (a3,d2.w),d5 move.w d5, pz move.w (al,d4.w),d5 * Calculate vector Q sub. w (al,d2.w),d5 move.w (a2,d4.w),d6 sub. w (a2,d2.w),d6 move.w (a3,d4.w),d7 sub. w (a3,d2.w),d7 230 Abacus Software ST 3D Graphics move. w d5,dl * qx move.w d6,d2 * qy move.w d7,d3 * qz mu Is py,d3 * Compute cross product mu Is pz, d2 * of the vector perpendicular sub.w d2,d3 * to the current surface move. w d3, rx mu Is pz, dl mu Is px,d7 sub.w d7, dl * The direction coordinates of move.w dl, ry * the vector perpendicular to mu Is px, d6 * the surface are stored mu Is py,d5 * in rx,ry,rz sub - w d5,d6 move. w d6, rz move. w prox,dl * The projection center serves as sub.w kx, dl * comparison point for the visibility move .w proy,d2 * of a surface which seems sub.w ky,d2 * adquate for the viewing move. w proz,d3 * situation. The observation sub.w kz, d3 * ref. point can also mu Is rx, dl * be used as the comparison point. muls ry,d2 * Compare vector R and mu Is rz, d3 * the vector from one add. 1 dl,d2 * point of the surface to add. 1 d2,d3 * the projection center by forming bmi dosight * the scalar product of the two vector: scalar product is negative, surface is visible move.w (aO),dl * Number of lines in surface ext. 1 dl lsl.l #2, dl ★ Number of lines times 4 = space for line addq.1 #2, dl * plus 2 bytes for number of lines add. 1 dl, aO ★ add to surface array for dbra dO,visible * access to next surface bra bideend * All surfaces processed ? End 231 ST 3D Graphics Abacus Software dosight: move.w (aO),dl * Number of lines for this surface ext.1 dl * multiplied by two results in ************* ** Changes from the program rotl.s * * ********************************** move.1 dl,d2 lsl.l #l,dl move.l a0,a4 addq.l #2,a4 move.w #0,zsurf sight2: move.l (a4)+,d6 swap d6 subq.w #l,d6 lsl.w #1,d6 move.w (a3,d6.w),d6 add.w d6,zsurf dbra d2,sight2 move.w zsurf,d6 ★ ext.l d6 lsr.l #2,d6 ext.l d6 move.1 d6, (a6) + move.1 aO, (a6) + sight3: move.w (a0)+, (a5) + dbra dl,sight3 addq.w #1,surfcount bra sight 1 hideend: rts * it * * * Number of words to be passed * Access to first line of the surface * Clear addition storage * first line of surface * first point in lower half of DO * fit index * fit operand size (2-Byte) * Z-coordinate of this point * add all Z-coordinates * until all lines are computed * Divide sum of all Z-coordinates * for this * surface by the number of lines * Surfaces created by rotation * always have four lines. * Store middle Z-Coordinate * followed by address of surface * pass number of lines * and individual lines * increase number of surfaces by one * and work on next surface 232 Abacus Software ST 3D Graphics *********************************************************************** * Create rotation body by passing parameters, * * rotating the definition line, and creating the line and * surface arrays * *********************************************************************** rlset: move.1 #rlxdat,rotxdat * Pass the move. 1 #rlydat,rotydat * parameters for move.1 #rlzdat,rotzdat * rotation body to move.1 #rldatx,rotdatx * routine for move.1 #rldaty,rotdaty * creating the move.1 #rldatz,rotdatz ★ rotation body move.1 rotdatx,datx * array addresses of move.1 move. 1 rotdaty,daty rotdatz,datz ★ the points move.w rlnumro,numro ★ Number of desired rotations move. w rlnumpt,numpt ★ Number of points to be rotated move.1 #rllin,linxy ★ Address of line array move.1 rts #rlplane,worldpla * Address of surface array rotstart: move.w numpt,dO lsl.w #l,dO ext.1 dO move.1 dO,plusrot move.w numpt,nummark move.1 rotdatx,pointx move.1 rotdaty,pointy move.1 rotdatz,pointz move.w #0,yangle move.w #360,dO divs numro,d0 move.w dO,plusagle move.w numro,d0 ext.l dO rloopl: move.1 d0,loopc * as loop counter move.l rotxdat,datx move.1 rotydat,daty move.l rotzdat,datz jsr yrot * rotate * Rotation of def line * numro+1 times about Y-axis * Storage for one line * Number of points * rotated * 360 / numro = angle increment * per rotation * store * numro +1 times 233 ST 3D Graphics Abacus Software rotlin: rotlopl: rotlop2: move.1 pointx,dl * add offset add. 1 plusrot,dl move.1 dl,pointx move.1 pointy, dl add. 1 plusrot,dl move. 1 dl,pointy move. 1 pointz,dl add. 1 plusrot,dl move.1 dl,pointz move.w yangle,d7 add.w plusagle,d7 move. w d7,yangle move.1 loopc,dO dbra dO,rloopl move.w rlnumro,numro move.w rlnumpt,numpt jsr rotlin * Create line array jsr rotsurf * Create surface array rts move.w #1, d7 move. w numro,d4 * * Number of rotations ext .1 d4 subq.1 #1, d4 move. w numpt,dl ★ Number of points in def. lin subq.w #1, dl * both as counters lsl. w #2,dl ★ times two ext. 1 dl move.1 dl,plusrot move.w numpt,d5 ★ Number of points minus once ext. 1 d5 * repeat, last line subq.1 #2,d5 ★ connect points (n-l,n) move.1 linxy,al move.w d7,d6 move.w d6, (al) + ★ first line connects addq.w #l,d6 ★ points (1,2) then (2,3) etc. move.w d6, (al) + dbra d5,rotlop2 234 Abacus Software ST 3D Graphics move.1 linxy,dl add. 1 plusrot,dl move . 1 dl,linxy move .w numpt,dO add. w d0,d7 dbra d4,rotlopl move . w numpt,d7 move.w d7,deltal lsl. w #2,d7 ext. 1 d7 move . 1 d7,plusrot move. w #1, d6 move.w numpt,dO ext. 1 dO subq.1 #l,dO rotlop3: move.w numro,dl ext. 1 dl subq.1 #1, dl move. w d6, d5 rotlop4 : move . w d5,(al)+ ★ generate cross add. w deltal,d5 * connection lines which move. w d5, (al) + * connect lines created dbra dl,rotlop4 ★ by rotation add.w #1, d6 dbra dO,rotlop3 move.w numro,dl add. w #1, dl mu Is nummark, dl move. w dl,rlnummark move.w numpt,dl mu Is numro,dl move. w numpt,d2 subq.w #1, d2 mu Is numro,d2 add.w dl, d2 235 ST 3D Graphics Abacus Software move.w d2,rlnumline * store number of lines rt s rotsurf: move.w numro,dO ext. 1 dO subq.1 #1, dO move.w numpt,d7 ext. 1 d7 subq.1 #2,d7 move.1 d7,plusrot move.1 worldpla,aO move.w #1/ dl move. w numpt,d2 addq.w #l,d2 rotf11: move.1 plusrot,d7 rotf12: move.w dl, d4 move.w d2,d5 addq.w #l,d4 addq.w #1, d5 move.w #4, (aO) + move.w dl, (aO) + move.w d4, (aO) + move.w d4, (aO) + move.w d5,(aO)+ move.w d5,(aO)+ move.w d2,(aO)+ move.w d2, (aO) + move.w dl, (aO) + addq.w #1, dl addq.w #1, d2 dbra d7,rot f12 addq.w #1, dl addq.w #1, d2 dbra dO,rotf11 move.w numpt, dl subq.w #1, dl muls numro,dl * Create surfaces of * rotation body * Number of points minus one * repeat * Address of surface array * Number of points * Offset * Number of lines/surfaces * first surface created here 236 Abacus Software ST 3D Graphics move.w dl,rlnumsurf rts * Pass data and parameters to the link file routines wrld2set: move.1 move -1 move.1 move.1 move.1 move . 1 move.1 move. w move. w move.w move.w move. 1 move. 1 move.w move.w move.w rts #wrldx,datx #wrldy,daty #wrldz,datz #viewx,pointx #viewy,pointy #viewz,pointz #wlinxy,linxy picturex,xO picturey, yO proz,zobs rlzl,dist #screenx,xplot #screeny,yplot rlnumline,numline rlnummark,nummark rlnumsurf,numsurf * Sort surfaces stored in pladress sort it: move. 1 #pladress,aO move. w surfcount,d7 ext. 1 d7 subq. 1 #2, d7 bmi serror move. 1 #l,dl sortmain: move . 1 dl, d2 subq.1 #l,d2 move.1 dl, d3 lsl.1 #3, d3 *********************************** ★ *********************************** * for i = 2 to n corresponds to * for i = 1 to n-1 because of * different array structure * j = i -1 * i 237 ST 3D Graphics Abacus Software move.1 (a0,d3.1) ,d5 * Comparison value x = a[i] move.1 4(aO,d3.1),d6 h Address of surface move. 1 d5,platz * a [0] = x = a [-1] in move.1 d6,platz+4 * this array sortlopl: move.1 d2, d4 * j lsl.l #3, d4 * j times 8 for access to array cmp. 1 (aO,d4.1),d5 * Z-coordinate of surface bge sortwl * while x < a[j] do move. 1 (aO, d4.1), 8(aO,d4 1.1) * a [ j + 1] = a [ j] move.1 4(aO,d4.1),12(aO, d4.1) * Address of surface ar: subq.1 #1, d2 * j = 3-1 bra sortlopl sortwl: move. 1 d5, 8 (aO, d4.1) * a Cj+1] = x move. 1 d6, 12(aO,d4.1) * pass address also addq. 1 #1, dl * i = i + 1 dbra d7, sortmain * until all surfaces are sorted sortend: rts serror: rts * On error simply return * Fill surfaces stored in pladress * paintit: jsr setclip * GEM clipping routine for Filled Area jsr sortit Hr Sort surfaces according to Z-coords. move.w #1, dO * Write mode to replace jsr filmode jsr filform * border filled surfaces jsr filcolor * Fill color is one move.w #2,dO ★ Fill style jsr filstyle move.1 xplot,al * Address of screen coordinates move.1 yplot,a2 move.w surfcount,d7 * Number of surfaces to be filled ext. 1 d7 Hr as counter subq.1 #1/ d7 * access to last surface in the array move.1 d7, dO * multiply by eight lsl.l #3, dO 238 Abacus Software ST 3D Graphics pa inti: paint2: paint3: move.1 #pladress,aO * here are the surfaces move.1 (a0,d0.1),d5 * largest Z-coordinate move. 1 #0, dl move. 1 (aO,dl.1), d6 * first surface in array neg. 1 d6 * smallest Z-coordinate add. 1 d6,d5 * subtract from each other move. 1 d5,d0 move.1 (a0,dl.l),d2 * first surface in array add. 1 d6, d2 * plus smallest Z-coordinate lsl.l #3,d2 * times eight, eight different divs d0,d2 * fill patterns, divide by difference neg.w d2 * leave out last pattern add. w #6,d2 bpl paint2 move.w #1, d2 move•w d2,d0 * Set fill index jsr filindex move. 1 #ptsin,a3 * Enter points here move. 1 4(a0,dl.l),a6 * Address of surface move. w (a6)+,d4 * Number of lines addq. w #1, d4 * first point counts double move. w d4,contrl+2 move. 1 (a6)+,d3 * first line of surface swap d3 subq.w #1, d3 lsl. w #1, d3 move.w (al,d3.w),(a3)+ * transfer to ptsin array move.w (a2, d3 . w) , (a3) + * transmit Y-coordinate swap d3 sub. w #l,d3 lsl. w #1, d3 move.w (al, d3 . w) , (a3) + * transmit next point move.w (a2,d3 .w) ,(a3)+ * transmit Y-coordinate subq.w #3,d4 * two points already transmitted ext. 1 d4 * one because of dbra move.1 (a6)+,d3 * next line subq.w #1, d3 lsl -W #1, d3 move.w (al,d3 .w) ,(a3)+ * X-coordinate move.w (a2,d3.w), (a3) + * Y-coordinate dbra d4,paint3 * until all points in ptsin arra; 239 ST 3D Graphics Abacus Software move.w #9,contrl * then call the move.w #0,contrl+6 * function Filled move. w grhandle,contrl+12 * Area movem.1 d0-d2/a0-a2,- (a7) jsr vdi movem.1 U7)+,d0-d2/a0-a2 add. 1 #8, dl * work on next dbra d7,paintl * surface in plad. rts * VDI clipping, used only with VDI functions, also for * * filling surfaces. * setclip: move.w #129,contrl move. w #2,contrl+2 move.w #1,contrl+6 move.w grhandle,contrl+12 move.w #1,intin move. w clipxule,ptsin move . w clipyule,ptsin+2 move. w clipxlri,ptsin+4 move.w clipylri,ptsin+6 jsr vdi rts . even * Start of variable area * * * * Data area for rotation body * .bss 240 Abacus Software ST 3D Graphics numro: numpt: rotxdat: rotydat: rotzdat: rotdatx: rotdaty: rotdatz: rlnumline: rlnummark: rlnumsurf: plusagle: rldatx: rldaty: rldatz: rllin: rlplane: rlxdat: rlydat: rlzdat: rlnumpt: rlnumro: .ds.w 1 .ds.w 1 . ds . 1 1 .ds.l 1 .ds.l 1 .ds.l 1 .ds.1 1 .ds.l 1 .ds.w 1 .ds.w 1 .ds.w 1 .ds.w 1 .ds.w 1540 .ds.w 1540 .ds.w 1540 •ds.l 3200 * 4-Bytes for every line .ds.l 6600 .data .dc.w 0,40,50,50,20,30,20,30,70,80,80,0 .dc.w 100,100, 80 ,60,40,30,30,-70,-80,-90,-100,-100 .dc.w 0 , 0 , 0 , 0 , 0 , 0,0,0,0, 0,0, 0 .dc.w 12 .dc.w 8 * Number of rotations for creation 241 ST 3D Graphics Abacus Software * * * * * Definition of the house * * * • data housdatx: .dc.w -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 . dc. w 30,30,30,30,30,30,30,30,30,30,30,30 housdaty: . dc. w 30,30,-30,-30,30,30,-30,-30,70,70,-30,0,0,-30 . dc. w 20,20,0,0,20,20,0,0 .dc. w -10,-10,-30,-30 housdatz: .dc .w 60,60, 60, 60,-60,-60,-60,-60, 60,-60, 60,60, 60, 60 .dc.w 40,10,10,40,-10,-40,-40,-10 .dc. w 0,-20,-20,0 houslin: .dc.w 1,2,2,3,3, 4,4,1,2,5, 5, 8,8, 3, 8,7, 7, 6, 6, 5, 6,1,7, 4 .dc.w 9,10,1,9,9,2,5,10,6,10,11,12,12,13,13,14 .dc.w 15,16, 16, 17,17,18; 18, 15,19,20,20,21,21,22,22, 19 .dc.w 23,24,24,25,25,26,26,23 * here are the definitions of the surfaces for the House * * * houspla: .dc. w 4, 1 ,2,2,3,3,4,4, 1,4,2,5,5,8, 8,3, 3,2 .dc. w 4, 5,6, 6,7,7,8,8, 5, 4,7, 6, 6,1, 1*4, 4,7 .dc. w 4, 4,3,3,8,8,7,7, 4,4,2, 9, 9,10 , 10, 5,5, 2 .dc. w 4, 10,9,9,1,1,6,6 ,10,3,1, 9, 9, 2 ,2, 1 .dc. w 3, 5,10,10,6,6,5, 4,11,12,12,13,13 *14, 14,11 .dc. w 4, 15,16,16,17,17 ,18,18,15,4, 19,2 0,20 ,21,21,22,22,19 -dc. w 4, 23,24,24,25,25 ,26,26,23 hnuinmark: .dc. w 26 * Number of corner points in the house hnumline: . dc. w 32 * Number of lines in the house hnumsurf: .dc :. w 13 * Number of surfaces in the house 242 Abacus Software ST 3D Graphics hxangle: .dc.w 0 ★ Rotation angle of house about X-axis hyangle: . dc. w 0 ★ " ” " Y-axis hzangle: .dc.w 0 * ■' " " Z-axis xwplus: .dc.w 0 ★ Angle increment about X-axis ywplus: • dc.w 0 ★ Angle increment about Y-axis zwplus: • dc.w 0 ★ Angle increment about Z-axis picturex: .dc.w 0 * Definition of zero point of display picturey: .dc.w 0 * entered by getreso rotdpx: .dc.w 0 rotdpy: .dc.w 0 rotdpz: .dc.w 0 rlzl: .dc.w 0 normz: .dc.w 1 .bss plusrot: .ds.l 1 first: • ds .w 1 second: .ds. w 1 deltal: .ds. w 1 worldpla: .ds.l 1 .data plag: .dc .b 1 . even .bss diffz: .ds. w 1 243 ST 3D Graphics Abacus Software dx: . ds. w 1 dy: . ds. w 1 dz: . ds. w 1 wrldx: .ds .w 1600 * World coordinate array wrldy: .ds. w 1600 wrldz: • ds. w 1600 viewx: .ds. w 1600 * View coordinate array viewy: • ds. w 1600 viewz: . ds. w 1600 screenx: .ds .w 1600 * Screen coordinate array screeny: .ds. w 1600 wlinxy: -ds. 1 3200 * Line array wplane: • ds. 1 6600 * Surface array vplane: -ds. 1 6600 * Surface array of visible surfaces platz: .ds. 1 2 pladress: • ds.l 3000 * Surface array surfcount: .ds .w 1 numsurf: . ds. w 1 zcount: .ds.l 1 * Sum of all Z-coord. zsurf: • ds. w 1 * Individual Z-coord.of surface sx: -ds. w 1 sy: .ds. w 1 sz : .ds .w 1 px: -ds. w 1 py: • ds. w 1 pz: .ds. w 1 244 Abacus Software ST 3D Graphics rx: .ds. w 1 ry: . ds. w 1 rz: . ds. w 1 qx: .ds. w 1 qy: .ds. w 1 qz: .ds.w 1 kx: . ds. w 1 ky: • ds.w 1 kz: • ds.w 1 . data prox: .dc.w 0 * Coordinates of projection proy: .dc. w 0 * center, on the positive proz: • dc.w 1500 * Z-axis .data of f x: .dc.w 0 * Transformation through rotation of fy: • dc.w 0 * to point [offx,offy,offz] of fz: .dc.w 0 xoffs: .dc.w 0 * Inverse transformation to point yoffs: .dc.w 0 * [xoff,yoffs,zoffs] zoffs: .dc.w 0 .bss loopc: .ds. 1 . end 1 245 ST 3D Graphics Abacus Software 4.4.1 New things in the main program rotatel.s: The creation of a surface array during construction of the rotation body is accomplished through the subroutine rotsurf:. The array (rlplane) is of course passed from the subroutine makewrld: into the world system (wplane). Furthermore, the subroutines hideit:, setclip: and paintit: as well as the sort routine sort it: are new and have already been explained. This sort routine sorts the array surfaddr, which contains the Z-coordinates of the visible surfaces as well as the addresses of the visible surfaces, according to increasing Z-coordinates. The subroutine sort it: uses the old trick, an additional array index at the beginning of the array. You can recognize this by the variable space: in the variable part of the program. The variable space: reserves additional space for a data record in the surfaddr-arrays. The additional space is used as a marker during sorting. The actual sort algorithm is nothing but a simple insert sort. For better understanding, here is a structogram of the sort algorithm: Figure 4.4.4: Structogram of the sort algorithm 246 Abacus Software ST 3D Graphics 4.4.2 Sort algorithm: In this program too, you should change various parameters to see what they do. Up to now you had to change all the parameters in the program text. This meant that you had to do a lot of assembling and linking just to change a few parameters. The sort algorithm will allow you to change parameters while the program is running. One method to change these parameters is through a menu. See the diagram below. More about this in the next section. ST 3D Graphics Abacus Software 4.5 Entering rotation lines with the mouse We are now ready to combine the subroutines which we have so far used separately and to construct a little program for creating rotation bodies, including the removal of hidden lines and shading surfaces. Furthermore, we also want to be able to enter the creation lines for the rotation body with the mouse so that we don’t have to reassemble the program when we want to use a new definition line. Owners of 520ST’s may find themselves running short of memory. The available storage space permits the input of 25 points for a definition line of the rotation body which can then be rotated 60 times about the Y-axis. Thus a maximum of 25*61 = 1525 points and about 3000 lines and almost 1500 surfaces will be created. To store this many parameters as well as the program we need about 190Kbytes of memory, about a third of which is wasted because the object is defined twice (datx, daty, datz, wrldx, wrldy, wrldz). This is done to make things easier, but also in consideration of the next main program which displays several objects at the same time. We also have to keep in mind the memory require by the two screen pages-about 64K The amount of memory reserved in this program is intended for use on the "smaller" model. Owners of 1 mega byte computers can display larger objects if they want by reserving more space for the individual arrays. The following relationships as are used to calculate the memory requirements: Number of points:= rlnumpt * (rlnumo+1) Number of lines:= ( (rlnumpt-1) *rlnumro) + (r1numpt * r1nuraro) Number of surfaces:= (rlnumpt-1) * rlnumro The number of lines can be estimated by multiplying the number of points by two. Each point naturally requires two bytes of storage space. You must also remember that every surface of the rotation body, requires 18 bytes of storage space since it is always constructed of four lines. In the surfaddr array every surface requires 8 bytes of additional storage space. With this information you can expand the programs yourself if you have a 1040ST. The introduction of the operating system in ROM will ease the lack of storage space. About 200K of RAM will be released by 248 Abacus Software ST 3D Graphics using the ROM. If you want to generate rotation bodies with more points without RAM enhancement, whether through ROMs or RAM chips, you can change the program so that the rotation body is not duplicated in the arrays rldatx, rldaty, rldatz, but generated only in the world system wrldx, wrldy, wrldz and the definition of rldatx, rldaty, rldatz is completely omitted. This will free about 50 Kbytes of storage which includes the savings from the line array (rllin) and surface array (rlplane). This space can be distributed over the world array and thus used to generate larger bodies. The product of the number of points and the number of rotations plus one is limited. You can for example, rotate 16 points 90 times, or 40 points 30 times, etc. The only limits placed are those of your imagination. The number of rotation points to be entered is determined by the variable maxpoint and can be changed there. The use of this program differs in a few points from the programs presented thus far. After the program start, a menu appears where you can determine the desired number of rotations of a rotation line already defined in the program. After you press one of the function keys F2 to F8, the familiar chess figure appears in the ’’wire model mode" with the desired number of rotations. The actual rotation parameters such as position of the rotation point and rotation angle increments can be changed with the cursor-keys. To remove hidden lines in this rotation body press the H key on the keyboard (H for Hide). After the visible surfaces have been drawn, you can fill them with a pattern by pressing the P key (P for Paint). In both cases you can obtain a hardcopy by pressing the <Altemate> and <Help> keys at the same time since the surfaces are drawn and in the visible screen page (physical display ). The picture drawn on the display remains until the <Retum> key is pressed and cannot be changed. As a further option you can fill all the surfaces in the "wire model mode" (P key), not only the visible ones. For hardcopy of a wire model, press Shift D. By pressing the F10 key you return to the main menu and you can enter a new rotation line with FI and the help of the mouse. After pressing FI a small crosshair and a cartesian coordinate system whose origin is the. middle of the screen appear. By clicking the left mouse button you can enter up to 25 points for a definition line. The right mouse button ends the definition after which you must press a key to return to the menu. You can set the number of rotations with the function keys. We almost forgot to mention the significance of the F9 function key which displays a mouse pointer when pressed in the wire model mode 249 ST 3D Graphics Abacus Software and allows you to set a new coordinate origin on the screen (left mouse button). Here are some examples of definition lines and the rotation bodies which result. Abacus Software ST 3D Graphics ST 3D Graphics Abacus Software 252 Abacus Software ST 3D Graphics ST 3D Graphics Abacus Software Abacus Software ST 3D Graphics ST 3D Graphics Abacus Software Figure 4.5.13 Don’t let the program listing frighten you. First of all, if you have entered the previous programs, all you have to do is enter the new subroutines and change the main loop a bit. Second, you can get a disk containing all of the programs in the book from Abacus Software or your dealer. 253882 bytes used 1 K PRINTERS N TUTORIAL C FRY CONV TTP HL10 PRO OUTPUT PRG SPLIT TTP STANDARD PRT TEXTPRO PRG TUTORIAL TXT XTTUTORI TOC l 1442236 bytes used in 123 itens. BASIC PRG 138344 11-2 BASIC RSC 4648 11-2 BASIC URK 346 11-2 BASIC1 BAR 14801 11-2 □ OPEN APPLICATION Nane: BATCH .TTP Parameters: . aslink grlinkl nenul|_ 1 333356 butes used in H0USE1 PRG H0USE1 S MAINi PR6 HAIN1 HAIN1C0 S PRG MAIN1CD S MENU1 PRG MENU1 S HULTI1 PRG _ HULTI1 S PAINT1 PRG ■ PAIKT1 R0TATE1 S PRG 1 R0TATE1 S ■ Abacus Software ST 3D Graphics *********************************************************************** * menul.s 2/18/1986 * * Creation of rotation bodies Uwe Braun 1985 Version 2.2 * * with hidden line algorithm and painting * * * .globl main,xoffs,yoffs,zoffs,offx,offy,offz •globl viewx,viewy,viewz .globl wlinxy,mouse_off,setrotdp,inp_chan,pointrot . text main: mainl: mainlopl: jsr apinit * Announce programm jsr grafhand * Get screen handler jsr openwork * Display jsr mouse off * Turn off mouse jsr getreso * Display resolution jsr setcocli * set Cohen Sutherland clip. jsr clearbuf jsr menu jsr makerotl * create rotation body jsr makewrld * create world system jsr wrld2set * pass world parameters jsr pageup jsr clwork jsr setrotdp * initialize observer ref. point jsr pagedown * Display logical screen page jsr clwork jsr inp_chan jsr pointrot * rotate around observ. ref. point jsr pers * Perspective transformation jsr drawnl jsr pageup * Display physical screen page jsr testhide 257 ST 3D Graphics Abacus Software jsr inp chan * Input new parameters jsr clwork * clear page not displayed jsr pointrot * Rotate around rot ref . point jsr pers * Transform new points jsr drawnl jsr pagedown * Display this logical page jsr inp chan * Input and change parameters jsr clwork * erase physical page jmp mainlopl * to main loop move.1 physbase,logbase jsr pageup * switch to normal screen page rts * back to link file and end * Display menu and selection of menu points menu: jsr switch * Display and draw the same move.1 #text2,a0 * screen page jsr printf * Display menu list move.1 #text3,aO jsr printf menuO: jsr inkey * Read keyboard swap dO cmp.b #$3b,dO * Fl key pressed ? bne menul jsr inpmous * if yes, enter a line bra menu menul: cmp .b #$3c,d0 * F2 key pressed ? bne menu 2 move.w #4, rlnumro * if yes, then initial number of bra menend * rotations to four 258 Abacus Software ST 3D Graphics menu2: cmp.b #$3d,d0 * F3 key bne menu 3 move.w #8,rlnumro bra menend menu3: cmp.b #$3e,d0 * F4 key bne menu 4 move.w #12,rlnumro bra menend menu*?: cmp.b #$3f,dO * F5 key bne menu 5 move.w #18, rlnumro bra menend menu5: cmp.b #$40,dO * F6 key bne menu6 move. w #24,rlnumro bra menend menu6: cmp.b #$41,dO * F7 key bne menu7 move. w #45,rlnumro bra menend menu7: cmp.b #$42,dO * F8 key bne menu 8 move.w #60,rlnumro bra menend menu8 : * Room for menu9: cmp. b #$44,dO * F10 key bne menuO addq.1 #4,a7 bra ma inend menend: rts additional keyboard commands 259 ST 3D Graphics Abacus Software ***************************************************************** * Test if removal of hidden surface and shading of surfaces * * is desired * ***************************************************************** jsr inkey * Read keyboard swap dO cmp.b #$23,dO ★ h key pressed ? beq dohide * if yes, call hideit cmp .b #$19,dO * p key pressed ? beq dopaint * is yes, shade rts ★ if not, return ★A***************************************************************** * Call hideit routine to remove hidden Surfaces * ******************************************************************* dohide: jsr switch jsr clwork jsr hideit jsr surfdraw dohidel: jsr inkey swap dO cmp.b #$19,dO beq dopain2 cmp.b #$lc,d0 bne dohidel jsr rts pageup dopain2: jsr paintit dopain3: jsr inkey swap dO cmp.b #$lc,dO bne dopain3 jsr rts pageup * or you won't see anything * erase display * remove * and draw * shade too ? * if yes, call fill routine * if not, wait for activation of * Return key on main keyboard * and back * Shade surfaces * wait for return key 260 Abacus Software ST 3D Graphics *********************************************************************** * Shade all surfaces defined in the world system **************,******************************************************** dopaint: jsr switch jsr clwork jsr paintall * shade all dopaintl: jsr inkey swap dO cmp. b #$lc,dO * and wait for Return key on the bne dopaintl * main keyboard jsr pageup rts * Create the rotation body makerot1: jsr jsr rts r lset * Set parameters of this rot. body rotstart * Create rot. body * Input and change parameters inp chan: jsr inkey cmp.b #' D',dO bne inpwait jsr scrdmp inpwait: swap dO cmp.b #$4d,d0 bne inpl addq.w #1,ywplus bra inpendl * Read keyboard, key code in * Make hardcopy * Test DO for * Cursor-right * if yes, add one to Y-angle * and continue 261 ST 3D Graphics Abacus Software inpl: cmp .b #$4b,dO bne inp2 subq.w #1,ywplus bra inpendl inp2: cmp .b #$50,dO bne inp3 addq.w #1, xwplus bra inpendl inp3: cmp .b #$48,dO bne inp3a subq.w #1,xwplus bra inpendl inp3a: cmp.b #$61,dO bne inp3b subq.w #1,zwplus bra inpendl inp3b: cmp.b #$62,dO bne inp4 addq.w #1,zwplus bra inpendl inp4: cmp.b #$4e,dO bne inp5 sub. w #25,dist bra inpendl inp5: cmp.b #$4a,d0 bne inp6 add. w #25,dist bra inpendl inp6: cmp.b #$66,dO bne inp7 sub.w #15,rotdpz bra inpendl * Cursor-left, if yes, subtract * one from Y-angle increment * Cursor-down, if yes * add one to X-angle increment * Cursor-up * subtract one * Undo key * Help key * plus key on the keypad * if yes, subtract 25 from base of * projection plane (Z-coordinate) * minus key on the keypad * * if yes, add 25 * * key on keypad * if yes, subtract 15 from rotation * point Z-coordinate * make changes 262 Abacus Software ST 3D Graphics inp7 : cmp -b #$65,dO * Division key on keypad bne inp8 add. w #15,rotdpz * add 15 bra inpendl inp8: cmp. b #$43,dO * F9 pressed ?, if yes, bne inplO jsr newmidd * display new screen center bra inpendl inplO: cmp -b #$44,dO * F10 pressed ? bne inpendl addq.1 #4,a7 * if yes, jump to new input bra mainl inpendl: move.w hyangle,dl * Rotation angle about the Y-axis add.w ywplus,dl * add increment cmp. w #360,dl * if larger than 360, subtract 360 bge inpend2 cmp. w #-360,dl * if smaller than 360, ble inpend3 * add 360 bra inpend4 inpend2: sub. w #360,dl bra inpend4 inpend3: add. w #360,dl inpendl: move.w dl,hyangle move.w hxangle,dl * proceed in the same manner with add. w xwplus,dl * rotation angle about the X-axis cmp. w #360,dl bge inpend5 cmp. w #-360,dl ble inpend6 bra inpend7 inpend5: sub. w #360,dl bra inpend7 inpend6: add.w #360,dl inpend7: move . w dl,hxangle ★ 263 ST 3D Graphics Abacus Software move.w hzangle,dl add. w zwplus,dl cmp. w #360,dl bge inpend8 cmp. w #-360,dl ble inpend9 bra inpendlO inpend8: sub. w #360,dl bra inpendlO inpend9: add. w #360,dl inpendlO: move.w rt s dl,hzangle fr***************************************************************** * Set the location of the coordinate origin of the screen * * system with the mouse * ****************************************************************** newmidd: jsr switch jsr mousform ★ change mouse form newmiddl: move. w x0,d2 move. w y0,d3 jsr mouspos * wait for mouse input move. w x0,d2 ★ must be called for unknown reasons move. w y0,d3 ★ twice for one input of the jsr mouspos * Position cmp.b #$20,dl * left button ? if not, then bne newmiddl ★ once more from the beginning move. w a rv> *• X o * store new coordinates move.w d3,y0 rts ******************************************************************** * Determine the current screen resolution * ********■*********★**********★**************"****'****•***'*■*******■**** getreso: move.w #4,-(a7) trap #14 addq.l #2,a7 264 Abacus Software ST 3D Graphics cmp.w #2, dO bne getrl move.w #320, picturex * Monochrome monitor move. w #200,picturey bra getrend getrl: cmp. w #1, d0 bne getr2 move.w #320,picturex * medium resolution (640*200 move. w #100,picturey bra getrend getr2: move. w #160,picturex * low resolution (320*200) move. w #100,picturey getrend: rts * Hardcopy of screen, called by inp_chan scrdmp: move.w #20,-(al) trap #14 addq.l #2,a7 jsr clearbuf rts * Initialize the rotation reference point to [0,0,0] * move . w #0,dl * set the initial rotation move.w dl, rotdpx * ref. point move.w dl, rotdpy move. w dl,rotdpz move.w #0,hyangle * initial rotation angle move.w #0,hzangle move.w #0,hxangle move. w #0,ywplus move. w #0,xwplus move -w rts #0,zwplus 265 ST 3D Graphics Abacus Software * Rotation around the rot. ref. point about all three axes * move.w hxangle,xangle ★ rotate the world around the move. w hyangle,yangle move. w hzangle,zangle move.w rotdpx,dO ★ rotation ref. point move.w rotdpy,dl move.w rotdpz,d2 move. w dO,xoffs * add for inverse transformation move.w dl,yoffs move.w d2,zoffs neg. w dO neg. w dl neg.w d2 move.w dO,offx * subtract for tranformation move.w dl,offy move. w d2,offz jsr matinit ★ initialize matrix jsr zrotate ★ rotate 'matrix' about Z-axis jsr yrotate * rotate 'matrix' about Y-axis jsr xrotate * then rotate about X-axis jsr rotate A multiply point with matrix rts ********************************************************************* * Set the limit of display window for the Cohen-Sutherland clip * * algorithm built into the draw-line algorithm * * The limits are freely selectable by the user which makes the * * draw-line algorithm very flexible. * ********************************************************************* move. w #0,clipxule move.w #0,clipyule move. w picturex,dl Isl. w #1, dl * times two subq.w #1, dl ■k minus one equals move. w dl,clipxlri * 639 for monochrome move.w picturey,dl lsl. w #l,dl ★ times two minus one equals subq.w #1, dl * 399 for monochrome 266 Abacus Software ST 3D Graphics move.w dl,clipylri rt s * Transfer object data into the world system . makewrld: move.1 trldatx,al move. 1 trldaty,a2 move.1 #rldatz,a3 move. 1 #wrldx,a4 move. 1 #wrldy,a5 move . 1 #wrldz,a6 move. w rlnummark,dO ext. 1 dO subq. 1 #l,dO makewll: move.w (al) +, <a4) + move.w (a2) +, (a5) + move.w (a3)+,(a6)+ dbra dO,makewll move.w rlnumline,dO ext. 1 dO subq.1 #1, dO move.1 #rllin,al move. 1 twlinxy,a2 makewl2: move. 1 (al) +, (a2) + dbra d0,makewl2 move.1 worldpla,aO move.1 #wplane,al move.w rlnumsurf,dO ext. 1 dO subq.1 #1, dO makewld: move. w (aO) +,dl move. w dl, (al) + ext. 1 dl subq. 1 #1, dl makewl4: move . 1 (aO)+,(al)+ dbra dl, makew!4 * create the world system through * copying the point coordinates * into the world system * Number of lines * Copy lines into world Line * array * Adress of surface definition * of the body, * Number of surfaces on the body * as counter * All lines in this surface, * and of course the number of * surfaces copied to world surface * array * copy every line of this surface * to the world array 267 ST 3D Graphics Abacus Software wrldset: dbra d0,makewl3 * until all surfaces are completed rts move. 1 #wrldx,datx * Pass variables for move.1 #wrldy,daty * the rotation routine move.1 #wrldz,datz move.1 #viewx, pointx move.1 #viewy,pointy move -1 fviewz,pointz move.1 Iwlinxy,linxy move.w picturex,xO * Coordinate source for the move.w picturey,yO * screen system move.w proz,zobs * projection center move.w rlzl,dist * position of projection plane move.1 tscreenx,xplot move.1 iscreeny,yplot move, w hnumline,numline move, w hnummark,nummark move.w hnumsurf,numsurf rts 268 Abacus Software ST 3D Graphics ********************************************************************* * Enter visible surface into the vplane array ********************************************************************* hideit: visible: move . w numsurf, dO * Number of surfaces as counter ext. 1 dO subq.1 #1* dO move.1 #viewx,al * point coordinates stored here move . 1 #viewy,a2 move.1 ffviewz, a3 move.1 #wplane,aO ★ here is information for every move. 1 #vplane,a5 ★ surface move.w #0,surfcount * counts the known visible surfaces move.1 #pladress,a6 * Address of the surface storage move.w (aO),dl ★ start with first surface, number ext. 1 dl * of points in this surface in Dl move.w 2(aO),d2 * Offset of first point of this surface move.w 4(aO),.d3 * Offset of second point move. w 8(aO),d4 ★ Offset of third point subq.w #1, d2 * Subtract one from current point offset subq. w #1, d3 ★ for access to point srray subq.w #l,d4 lsl. w #lf d2 ★ then multiply by two lsl. w #l,d3 lsl. w #l,d4 * and finally access the current move. w (al,d3.w),d6 * point coordinates cmp. w (al,d4.w),d6 ★ comparison recognizes two points bne doitl ★ with some coordinates which can occur move. w (a2,d3.w) ,d6 ★ during construction of rotation cmp. w (a2,d4.w),d6 * bodies. If two bne doitl ★ points where all point coordinates move.w (a3,d4.w),d6 ★ (x,y,z) match, the program selects cmp.w <a3,d3.w),d6 * a third point to determine the two bne doitl ★ vectors move . w 12 (aO) , d4 subq.w #l,d4 lsl. w #1, d4 269 ST 3D Graphics Abacus Software doit1: * ★ move.w (al,d3.w) ,d5 * here the two vectors which lie in move.w d5, kx * surface plane are detemined by * subtraction sub. w (al, d2.w),d5 * of coordinates from two points of move.w d5, px * points in this surface move.w (a2,d3.w),d5 move.w d5, ky * the direction coordinates of the sub. w (a2,d2.w),d5 * vector are stored in the variable; move.w d5,py * qx,qy,qz and px,py,pz move. w (a3, d3 . w) , d5 move.w d5, kz sub. w (a3,d2.w),d5 move.w d5,pz move.w (al,d4.w),d5 * calculation of vector Q sub. w <al,d2.w),d5 move.w (a2,d4.w),d6 sub. w (a2,d2.w),d6 move. w {a3, d4.w),d7 sub. w (a3, d2.w),d7 move. w d5, dl * qx move.w d6,d2 * qy move. w d7,d3 * qz mu Is py,d3 * calculation of the cross product mu Is pZ/ d2 * of the vector perpendicular to * the surface sub.w d2,d3 move.w d3, rx muls pz, dl mu Is px, d7 sub. w d7, dl * the direction coordinates of * the vector move.w dl, ry * which is perpendicular to the muls px, d6 * surface area stored temporarily muls py,d5 * rx,ry,rz sub. w d5,d6 move.w d6, rz move.w prox,dl * The projection center is used as sub.w kx, dl * the comparison point for the 270 Abacus Software ST 3D Graphics move.w proy,d2 ★ visibility of a surface, which is sub. w Jcy,d2 ★ adequate for this viewing move. w proz,d3 * situation. One can also use sub.w kz, d3 * the observation ref. point mu Is rx, dl * as the comparison point. mu Is ry,d2 * Now follows the comparison of the muls rz, d3 * vector R and the vector from add. 1 dl, d2 * one point on the surface to the add. 1 d2,d3 * projection center by creating the bmi dosight ★ scalar product of the two vectors * the surface is visible, otherwise continue with next surface. move.w (aO),dl * Number of lines in surface ext.1 dl lsl.l #2,dl * Number of lines times 4 = space for lines addq.l #2,dl * plus 2 bytes for the number of lines add.1 dl,aO * add to surface array, for access to sight1: dbra dO,visible * next surface. If all surfaces bra hideend * completed, go to end. dosight: move.w <aO),dl * Number of lines in this surface ext.l dl * multiplied by two gives result of move.l dl,d2 lsl.l #l,dl * number of words to be transmitted move.l a0,a4 addq.l #2,a4 * Access to first line of surface move.w #0,zsurf * Erase addition storage (a4)+,d6 * first line of surface d6 * first point in lower half of DO #l,d6 * adapt Index #1,d6 * adapt Operand size (2-byte) move.w (a3,d6.w),d6 * Z-coordinate of this point add.w d6,zsurf * add all Z-Coordinates dbra d2,sight2 * until all lines have been processed sight2: move.l swap subq.w lsl. w 271 ST 3D Graphics Abacus Software move. w zsurf,d6 ★ ext. 1 d6 * lsr. 1 #2,d6 * ext .1 d6 * move. 1 d6, (a6) + * move.1 aO, <a6)+ * sight3: move.w (aO)+, (a5) + * dbra dl,sight3 * addq. w #1,surfcount * bra sight1 * hideend: rts Divide sum of all Z-coordinates of this surface by the number of lines in the surface. Surfaces created by rotation always have four lines store middle Z-coordinates followed by address of surface transmit the number of lines and the individual lines add one to the number of surfaces and work on next one * Draw all surfaces contained in vplane * surfdraw: * Draws the number of surfaces passed move.1 xplot,a4 ★ in surfcount whose descriptions move.1 yplot,a5 move.1 #vplane,a 6 ★ were entered by hideit in the array move.w surfcount,dO * at address vplane ext. 1 dO subq.1 #l,dO ★ if there are no surfaces in the arr. bmi surfend * then end. surflopl: move.w (a6) +,dl * Number of lines in this surface ext. 1 dl * as counter of lines to be drawn. subq.1 #1, dl surflop2: move.1 (a6)+,d5 ★ first line of this surface subq.w #1, d5 ★ Access to screen array where lsl. w #1, d5 ★ screen coordinates of points are. move. w 0 (a4,d5.w),d2 move . w 0 (a5,d5.w),d3 ★ extract points swap d5 * pass routine. 272 Abacus Software ST 3D Graphics subq. w #1, d5 lsl. w #1, d5 move.w 0 <a4, d5 . w) , a2 move.w 0(a5,d5.w),a3 jsr drawl dbra dl,surflop2 dbra dO,surflopl surfend: rts * second point belonging to * line * draw line, until all lines in this * surface are drawn and repeat * until all surfaces are drawn. * finally return. * Set parameters of this rotation body * rlset: move. 1 #rlxdat,rotxdat move. 1 #rlydat,rotydat move. 1 frlzdat,rotzdat move.1 #rldatx,rotdatx move.1 #rldaty,rotdaty move.1 #rldatz,rotdatz move . 1 rotdatx,datx move.1 rotdaty,daty move. 1 rotdatz,datz move.w rlnumro,numro move.w rlnumpt,numpt move.1 #rllin,linxy move.1 rts #rlplane,worldpla * Pass parameters of this * rotation body to routine * for generating the * rotation body * Array addresses of points * Number of desired rotatations. * Number of points to be rotated * Address of line array * Address of surface array ************** * and create rotation body •k ★ * rotstart: move.w lsl. w ext .1 move. 1 numpt,dO #1, dO dO dO,plusrot * Rotate the def line * numro+1 times about the Y-axis * Storage space for one line 273 ST 3D Graphics Abacus Software rloopl: rotlin: move.w numpt,nummark * Number of points move.1 rotdatx,pointx * rotate to here move. 1 rotdaty,pointy move. 1 rotdatz,pointz move. w #0, yangle move.w #360,dO * 360 / numro = angle increment divs numro,dO * per rotation move.w dO, plusagle * store move. w numro,dO * numro +1 times ext. 1 dO move.1 dO, loopc * as loop counter move.1 rotxdat,datx move.1 rotydat,daty move.1 rotzdat,datz jsr yrot * rotate move.1 pointx,dl * add offset add. 1 plusrot,dl move.1 dl,pointx move.1 pointy,dl add. 1 plusrot,dl move.1 dl,pointy move. 1 pointz,dl add. 1 plusrot,dl move.1 dl,pointz move. w yangle,d7 add.w plusagle,d7 move.w d7,yangle move.1 loopc,dO dbra dO,rloopl move.w rlnumro,numro move.w r1numpt,numpt jsr rotlin * Create line array jsr rotsurf * Create surface array rts move. w #l,d7 move.w numro,d4 * Number of rotations ext. 1 d4 subq.1 #l,d4 274 Abacus Software ST 3D Graphics rotlopl: rotlop2: rotlop3: rotlop4 : move. w numpt,dl * Number of points in the def. line subq.w #1, dl * both as counter lsl. w #2, dl * times two ext. 1 dl move. 1 dl,plusrot move. w numpt,d5 * Number of points minus one ext .1 d5 * repeat, last line subq.1 #2, d5 * connects the points (n-l,n) move.1 linxy,al move. w d7,d6 move.w d6, <al) + * the first line connects the addq.w #l,d6 * points (1,2) then (2,3) etc. move.w d6,(al)+ dbra d5,rotlop2 move.1 linxy,dl add. 1 plusrot,dl move.1 dl,linxy move.w numpt,dO add. w d0,d7 dbra d4,rotlopl move. w numpt,d7 move. w d7,deltal lsl. w #2, d7 ext. 1 d7 move. 1 d7,plusrot move. w #1, d6 move . w numpt,dO ext. 1 dO subq.1 #l,dO move. w numro,dl ext. 1 dl subq.1 #1, dl move.w d6,d5 move.w d5, (al) + * now generate the cross connections add. w deltal,d5 * which connect the individual lines move.w d5, (al) + * created by rotation dbra dl,rotlop4 275 ST 3D Graphics Abacus Software rotsurf: add. w #l,d6 dbra dO,rotlop3 move.w numro,dl add. w #l,dl mu Is nummark,dl move.w dl,rlnummark move.w numpt,dl mu Is numro,dl move.w numpt,d2 subq.w #1, d2 mu Is numro,d2 add. w dl, d2 move.w rts d2,rlnumline * Number of lines stored move. w numro,dO * create surfaces of the ext .1 dO * rotation body subq.1 #l,dO move. w numpt,d7 * Number of points minus one ext .1 d7 * repeat subq.1 #2,d7 move.1 d7,plusrot move.1 worldpla,aO * Address of surface array move.w #1, dl move.w numpt,d2 * Number of points addq.w #1, d2 276 Abacus Software ST 3D Graphics rotf11: rotf12 : move.l plusrot,d7 * Offset move.w dl,d4 move.w d2,d5 addq.w #l,d4 addq.w #l,d5 move.w #4, (aO> + * Number of lines / surfaces move.w dl,(aO)+ * the first surface is move.w d4,(a0) + * created here move. w d4, (aO) + move.w d5, (aO) + move . w d5, (aO) + move . w d2, (aO) + move . w d2, (aO) + move . w dl, (aO) + addq.w #l,dl addq.w #l,d2 dbra d7,rotfl2 addq.w #l,dl addq.w #l,d2 dbra dO,rotfll move.w numpt,dl subq.w #1, dl muls numro,dl move.w dl,rlnumsurf rts 277 ST 3D Graphics Abacus Software ************************************************************************ * Transfer the world parameters and the variables to the link file * ************************************************************************ wrld2set: move.1 move. 1 move.1 move.1 move.1 move.1 move.1 move.w move.w move.w move.w move.1 move.1 move.w move.w move -w rts #wrldx,datx * #wrldy,daty * #wrldz,datz * #viewx,pointx #viewy,pointy Iviewz,pointz #wlinxy,linxy picturex,xO picturey,yO proz,zobs rlzl,dist #screenx, xplot #screeny,yplot rlnumline,numline rlnummark,nummark rlnumsurf,numsurf transfer the world parameters and the variables to the routines in the link file ********************************************************************** * Sort all surfaces entered in pladress * sortit: sortmain: move.1 #pladress,aO move.w surfcount,d7 ext. 1 d7 •* for i = 2 to n corresponds to subq.1 #2, d7 * number of runs bmi serror ★ for i = 1 to n-1 because of move.1 #1, dl * different array structure move.1 dl, d2 subq.1 #l,d2 ★ j = i -1 move.1 dl, d3 * i lsl.l #3,d3 move.1 (aO, d3.1),d5 * Comparison value x = a[i] move.1 4(a0,d3.1),d6 * address of the surface move.1 d5,space * a [0] = x = a[-l] in this move.1 d6,space+4 * array 278 Abacus Software ST 3D Graphics sortlopl: move.1 d2,d4 * j lsl.l #3, d4 * j times 8 for access to array cmp. 1 (a0,d4.1) ,d5 * Z-coordinate of surface bge sortwl * while x < a[j] do move.1 (a0,d4.1),8(a0,d4 .1) * a [ j + 1] = a [ j] move.1 4(a0,d4.1),12(aO, d4.1) * Address of surface array subq.1 #1, d2 * j = j-1 bra sortlopl sortwl: move. 1 d5,8(aO,d4.1) * a[j+1] = x move. 1 d6,12(a0,d4.1) * Pass address also addq. 1 #1/ dl * i - i + 1 dbra d7,sortmain * Until all surfaces have been sorted sortend: rts serror: rts * On error simply return ******************************************************************** * paintall draws all surfaces in world array wplane independent of * * their visibility; all surface addresses and middle Z-coordinates * * are entered into the pladress array. * ********«**************+******************************************** paintall: move.w numsurf,dO ext.l dO subq.l #l,dO bmi pquit * Number of surfaces * if no surface present * then terminate move.1 #viewz,a3 move.l #wplane,aO move.w #0,surfcount move.l #pladress,a6 * Surface counter for surfdraw * surfaces are entered here svisible: move.w (aO),dl * all surfaces are visible ext.1 dl subq.l #l,dl move.w #0,zsurf * middle Z-coordinate move.l a0,a4 addq.l #2,a4 279 ST 3D Graphics Abacus Software ssightbl: move.1 (a4) +,d2 * first line of surface swap d2 subq.w #1» d2 lsl. w #1/ d2 ddoit1: move.w (a3,d2.w),d6 * add all Z-coordinates of this add. w d6,zsurf * surface dbra dl,ssightbl move.w zsurf,d6 ext. 1 d6 * then divide by four, shifting lsr. 1 #2,d6 * is possible only with rotation ext. 1 d6 * bodies since each surface has move.1 d6, <a6)+ * exactly four lines otherwise divide move.1 aO, (a6) + * by number of lines addq.w #1,surfcount * increment surface counter for surfdraw move.w (aO),dl * AO still points to number of lines ext. 1 dl * in this surface lsl.l #2, dl * Number of lines times four (1 long) addq.1 #2, dl * 2 bytes for the number of lines add. 1 dl, aO * AO points to next surface dbra dO, svisible move.w numsurf,surfcount jsr paintit * Fill surfaces in pladress pquit: rt s paintit: jsr setclip * GEM clipping routine for filled area jsr sortit * Sort surfaces according to Z-coordinates move.w #1, dO * Write mode to replace jsr filmode jsr filform * frame filled surface jsr filcolor * Shading color is one move. w #2, dO * Fill style jsr filstyle move. 1 xplot,al * Address of screen coordinates move. 1 yplot,a2 move. w surfcount, dl * Number of surface to be filled ext .1 d7 * as counter subq.1 #1, d? * access last surface in array move.1 dl , dO * multiply by eight 280 Abacus Software ST 3D Graphics lsl.l move. 1 move. 1 move.1 move.1 neg. 1 add. 1 paintl: move.1 move.1 add. 1 lsl.l divs neg.w add. w bpl move.w paint2: move.w jsr move.1 move.1 move.w addq.w move.w move.1 swap subq.w lsl. w move.w move.w swap sub. w lsl. w move.w move.w subq.w ext .1 paint3: move.1 subq.w lsl. w move.w move.w #3, dO tpladress,aO (aO, dO . 1) , d5 #0, dl (aO,dl.1), d6 d6 d6,d5 d5,d0 (aO,dl.l),d2 d6,d2 #3,d2 dO, d2 d2 #6,d2 paint2 #1, d2 d2, dO filindex #ptsin,a3 4 (aO, dl. 1) ,a6 (a6)+,d4 #1»<34 d4,contrl+2 (a6) + ,d3 d3 #l,d3 #l,d3 <al,d3.w>,(a3)+ (a2,d3.w) ,(a3) + d3 #1» d3 #l»d3 (al,d3.w),(a3)+ (a2, d3 . w) , (a3) + #3, d4 d4 (a6)+,d3 #1< d3 #l,d3 (al, d3 .w), (a3) + (a2, d3 . w), (a3) + * here are largest Z-coordinate * surfaces * first surface in array * smallest Z-coordinate * subtract from one another * first surface in array * plus smallest Z-coordinate * times eight, eight different * shading patterns, divide by * difference leave out last * pattern. * set fill index * enter points here * Address of surface * Number of lines * first point counted twice * first line of surface * transfer to ptsin array * pass Y-coordinate * transmit next point * transmit Y-coordinate * already two points transmitted * and one because of dbra * next line * X-coordinate * Y-coordinate 281 ST 3D Graphics Abacus Software dbra d4,paint3 * until all points in Ptsin-Array move.w #9,contrl * then call the fill area function move. w #0,contrl+6 move.w grhandle,contrl+12 movem.1 dO-d2/aO-a2,-(a7) jsr vdi movem.1 (a7)+,d0-d2/a0-a2 add. 1 #8, dl * work on next surface in pladress dbra d7,paintl rts * VDI clipping, only needed when VDI functions are used, * * for surface filling. setclip: move.w #129,contrl move.w #2,contrl+2 move.w #1,contrl+6 move.w grhandle,contrl+12 move.w #1,intin move.w clipxule,ptsin move.w clipyule,ptsin+2 move.w clipxlri,ptsin+4 move. w clipylri,ptsin+6 jsr vdi rts ******************************************************************* * this subroutine allows coordinates to entered with the Mouse * The maximum number of points is in the variable maxpoint, and * is limited only by storage space ******************************************************************* inpmous: jsr switch move. w #5, dO jsr setform move.w #1, dO * set input mode to mouse-reguest move.w #1, dl * wait for mouse input which is jsr setmode * terminated by key activation and 282 Abacus Software ST 3D Graphics mouslopl: mousl: mouslop2: jsr coord * mouse clicking move. 1 #0,adressx move.w #5,d0 * set polymarker to diagonal cross jsr marktype jsr mouspos * For unknown reasons function must move. w picturex,d2 * be called twice to work once. add. w #15,d2 move.w picturey,d3 sub.w #40,d3 jsr mouspos cmp .b #$20,dl * wait until the left mouse button is bne mouslopl * pressed move.1 #rlxdat,a4 * arrays in which input move.1 #rlydat,a5 * coordinates are entered; enough move.1 #rlzdat,a6 * storage must have been reserved move.w d2,newx * store mouse X and Y positions move.w d3,newy jsr saveit * and pass line array move.w newx,d2 move.w newy,d3 jsr markit * set a polymarker add. 1 #1,adressx * increment counter nop move.w newx,altx move.w newy,alty move.w altx,d2 * pass old position of the mouse move. w alty,d3 jsr mouspos * and call again jsr mouspos cmp .b #$21,dl * if right mouse button, then beq mousend * end of mouse input cmp.b #$20,dl bne mouslop2 move.w d2,newx * store mouse coordinates move.w d3,newy jsr saveit * store in array 283 ST 3D Graphics Abacus Software move.w newx,d2 * draw line from (n-1) n'th point move.w newy,d3 move.w altx,a2 move. w alty,a3 jsr drawl move.w newx,d2 move.w newy,d3 jsr markit * and mark point with marker add. 1 #1,adressx * increment counter move.1 adressx,d7 cmp. 1 maxpoint,d7 * and compare with maximum point count bne mousl * if not equal, continue move.1 adressx,dO move.w dO,rlnumpt * Number of points input rts mousend: move.w d2,newx move.w d3,newy move.w altx,a2 move.w alty,a3 jsr markit jsr drawl * draw last line jsr wait * and wait for keypress jsr saveit add. 1 #1,adressx * also add last point move. 1 adressx, dO move.w dO,rlnumpt * now store total number of points rts * finally back to caller *********************************************************************** * Wait for mouse input, returns also on keyboard input * *********************************************************************** mouspos: move.w #28,contrl * Mouse input, the desired coordinates move.w #1,contrl+2 * where the mouse should appear, move.w #0,contrl+6 * are passed in 284 Abacus Software ST 3D Graphics move. w grhandle,contrl+12 move . w d2,ptsin * D2 , and D3 move. w d3,ptsin+2 jsr vdi move.w intout,dl * the result - coordinates move. w ptsout,d2 * are also returned in D2 and move . w ptsout+2,d3 * D3 rts * Set the polymarker type * marktype: move.w move. w move. w move.w move.w jsr rts #18,contrl #0,contrl+2 #1,contrl+6 grhandle,contrl+12 dO,intin vdi * determines the appearance of * the polymarker, desired * type is passed in DO * Set a polymarker, number in contrl+2 * markit: move. w #7,contrl move. w #1,contrl+2 * Number of points. move.w #0,contrl+6 * case only one move.w grhandle,contrl+12 move. w d2,ptsin move.w d3,ptsin+2 movem.1 d0-d2/a0-a2,-(a7) jsr vdi * draw marker movem.1 (a7)+,dO-d2/aO-a2 rts 285 ST 3D Graphics Abacus Software A********************************************************************** * Set input mode *********************************************************************** setmode: move. w #33,contrl * Set input mode move.w #0,contrl+2 move.w #2,contrl+6 move.w grhandle,contrl+12 move.w dO,intin move.w dl,intin+2 * Parameters in DO jsr vdi rts * Store coordinates entered in point array * sub.w picturex,d2 * Pass mouse coordinates to move. w d2,(a4)+ * rotation line array, with sub.w picturey,d3 * adaptation to coordinate system neg.w d3 move.w d3, (a5) + move.w #0, (a6) + rts Abacus Software ST 3D Graphics ********************************************************************** * Display and describe the same screen page * ********************************************************************** move.w #-l,-(a7) * Display of Display Page, move.1 physbase. “ (a7) * where drawing is made move.1 physbase. - (a7) move.w #5,-(a7) trap #14 add. 1 #12,a7 rts * Change the mouse form * setform: move. w #78, contrl * Set mouse form move.w #1, contrl+2 move.w #1,contrl+4 * passed in DO move.w #1,contrl+6 move.w #0,contrl+8 move.w dO,intin jsr aes rts desired shape *********************************************************************** * Drawing a coordinate system for mouse input * *********************************************************************** jsr clwork. * draw coordinate move.w #0, d2 * for mouse input move.w picturey,d3 move.w picturex,d5 lsl. w #1, d5 move.w d5,a2 287 ST 3D Graphics Abacus Software move.w d3,a3 jsr drawl move.w picturex,d2 move.w #0, d3 move. w d2,a2 move.w picturey,d5 lsl. w #1, d5 move.w d5,a3 jsr drawl rts ****************************************************************** * remove all characters present in the keyboard buffer * ****************************************************************** clearbuf: clearnd: move - w #$b,-(a7) * Gemdos fnct. character in Buffer ? trap #1 addq.1 #2,a7 tst. w dO * if yes, get character beq clearnd * if no, terminate move.w #1,-U7> * Gemdos fnct. CONIN trap #1 * repeat, until all characters addq.1 #2,a7 * are removed from the buffer bra clearbuf rts * Definition of a custom mouse form - Data in mousforl * mousform: move.1 #15,dO * permits the definition of move.1 Imousforl,al * new mouse form, data is move.w #111,contrl * in mousforl move.w #0,contrl+2 move.w #37,contrl+6 move.w grhandle,contrl+12 move.w #8,intin move.w #8,intin+2 288 Abacus Software ST 3D Graphics move.w #1,intin+4 move.w #0,intin+6 move.w #1,intin+8 move.1 tintin+10,a5 move.1 (al)+,(a5)+ dbra dO,forlop jsr vdi rts .even * Beginning of the Variable area * * * * Data area for the rotation body * .bss numro: .ds. w 1 numpt: .ds. w 1 rotxdat: .ds.l 1 rotydat: .ds.l 1 rotzdat: .ds.l 1 rotdatx: .ds.l 1 rotdaty: .ds.l 1 rotdatz: .ds.l 1 rlnumline .ds. w 1 rlnummark .ds. w 1 rlnumsurf . ds. w 1 plusagle: .ds. w 1 rldatx: . ds. w 1600 rldaty: • ds. w 1600 rldatz: . ds . w 1600 289 ST 3D Graphics Abacus Software rllin: .ds.l 3200 * 4-Bytes for every line e rlplane: .ds.l 6600 • data rlxdat: .dc.w 0,40,50,50,20,30,20,30,70,80,80,0 .dc.w 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 rlydat: .dc.w 100,100,80,60,40,30,30,-70,-80,-90,-100,-100 .dc.w 0,0,0,0,0,0, 0,0, 0,0, 0,0,0,0, 0,0, 0,0, 0,0 rlzdat: .dc.w 0,0,0,0,0,0,0,0,0,0,0,0 .dc.w 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 rlnumpt: .dc.w 12 rlnumro: .dc.w 8 * * * * * Definition of the house * * * ************************************************************************ .data housdatx: .dc.w -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 .dc.w 30,30,30,30,30,30,30,30,30,30,30,30 housdaty: .dc.w 30,30,-30,-30,30,30,-30,-30,70,70,-30,0,0,-30 .dc.w 20,20,0,0,20,20,0,0 .dc.w -10,-10,-30,-30 housdatz: .dc.w 60 , 60 , 60 , 60 ,- 60 ,- 60 ,- 60 ,- 60 , 60 ,- 60 , 60 , 60 , 60,60 .dc.w 40,10,10,40,-10,-40,-40,-10 .dc.w 0,-20,-20,0 290 Abacus Software ST 3D Graphics houslin: .dc.w 1,2,2,3,3,4,4,1,2,5,5,8,8,3,8,7,7,6,6,5,6,1,7,4 .dc.w 9,10,1,9,9,2,5,10,6,10,11,12,12,13,13,14 .dc.w 15,16,16,17,17,18,18,15,19,20,20,21,21,22,22,19 .dc.w 23,24,24,25,25,26,26,23 *********************************************************************** * Here is the definition of the surfaces belonging to the house * *********************************************************************** houspla: .dc.w 4 , 1 , 2 , 2 , 3 , 3 ,4,4,1,4,2,5,5,8,8,3,3,2 .dc.w 4,5,6,6,7,7,8,8,5,4,7,6,6,1,1,4,4,7 .dc.w 4,4,3,3,8,8,7,7,4,4,2,9,9,10,10,5,5,2 .dc.w 4,10,9, 9,1,1, 6,6,10,3,1, 9, 9, 2,2,1 .dc.w 3,5,10,10,6,6,5,4,11,12,12,13,13,14,14,11 .dc.w 4 ,15,16,16,17,17,18,18,15,4,19,20,20,21,21,22,22,19 .dc.w 4,23,24,24,25,25,26,26,23 hnummark.: .dc.w 26 * Number of corner points in the house hnumline: .dc.w 32 * Number of Lines in the House hnumsurf: .dc.w 13 * Number of Surfaces in the House hxangle: .dc.w 0 * Rotation angle of House about the X-axis hyangle: .dc.w 0 * " " " Y-axis hzangle: .dc.w 0 * ” " " Z-Axis xwplus: .dc.w 0 * Angle increment about the X-axis ywplus: .dc.w 0 ★ Angle increment about the Y-axis zwplus: .dc.w 0 ★ Angle increment about the Z-axis picturex: .dc.w 0 ★ Definition of zero point of screen picturey: .dc.w 0 ★ entered by getreso rotdpx: .dc.w 0 rotdpy: .dc.w 0 rotdpz: .dc.w 0 291 ST 3D Graphics Abacus Software rlzl: . dc. w 0 normz: . dc. w 1500 .bss plusrot: . ds. 1 1 first: . ds. w 1 second: • ds. w 1 deltal: .ds. w 1 worldpla: •—i w T3 1 . data plag: • dc.b 1 .even .bss diffz: • ds .w 1 dx: .ds. w 1 dy: .ds. w 1 dz: • ds. w 1 wrldx: .ds. w 1600 * World coordinate array wrldy: . ds. w 1600 wrldz: . ds. w 1600 viewx: . ds. w 1600 * View coordinate array viewy: . ds. w 1600 viewz: . ds. w 1600 screenx: . ds. W 1600 * Screen coordinate array screeny: .ds .w 1600 292 Abacus Software ST 3D Graphics wlinxy: .ds. 1 3200 * Line array wplane: .ds. 1 6600 * Surface array vplane: .ds. 1 6600 * Surface array of visible surface space: .ds. 1 2 pladress: .ds. 1 3000 * Surface array surfcount : .ds.w 1 numsurf: .ds .w 1 zcount: .ds. 1 1 * Sum of all Z-coord. zsurf: .ds.w 1 * Individual Z-coord. of surface sx: .ds.w 1 sy: .ds.w 1 sz: .ds.w 1 px: .ds.w 1 py: .ds.w 1 pz: .ds.w 1 rx: .ds.w 1 ry: • ds.w 1 rz: • ds.w 1 qx: .ds.w 1 qy: .ds. w 1 qz: .ds.w 1 kx: .ds.w 1 ky: .ds.w 1 kz : .ds.w 1 293 ST 3D Graphics Abacus Software Hr ************** + ********************★*** .data • even maxpoint: .dc. 1 25 mousx: .dc.w 0 mousy: • dc.w 0 mousbut: .dc.w 0 kybdstat: .dc.w 0 altx: .dc.w 0 alty: .dc.w 0 newx: .dc.w 0 newy: .dc.w 0 adressx: .dc. 1 1 .data prox: .dc.w 0 * Coordinates of the projections proy: .dc.w 0 * center on the positive proz: .dc.w 1500 * Z-axis .data of fx: .dc.w 0 * Transformation during rotation of fy: .dc.w 0 * to point (offx,offy,offz] of f z: • dc.w 0 xoffs: • dc.w 0 * Inverse transformation to point yoffs: • dc.w 0 * [xoff,yoffs,zoffs] zoffs: .dc.w 0 textl: .dc.b 27,'Y',56,61,' <c) Uwe Braun 1985 ',0 text2: .dc .b 27,'E',27,'p',13,' Input ',' 4-Pts ' , ' .dc.b ' 12-Pts ' .dc.b ' 18-Pts 24-Pts 45-Pts 60-Pts .dc.b ' POS ',' Quit' ,27,'q',0 text3: .dc.b 13, 10,' F-l F-2 F-3 .dc.b ' F-5 F-6 F-7 F-8 .dc.b ' F-9 F-10 ' t 13,0 -Pts F-4 294 Abacus Software ST 3D Graphics mousforl: .dc.w .dc.w .dc.w • dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w mousdatl: .dc.w .dc.w .dc.w • dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w .dc.w • dc.w .dc.w .bss loopc: .ds.l . end %11111111111111U %iiiiiiiimniii %ooooooinuooooo %0000110000010000 %0001001111001000 %0010010000100100 %0100100000010010 %1001000000010100 %1001000000010100 %1000100000100101 %0100011111001001 %0010000000010010 %0001111111100101 %0011111111111001 %0111111111111111 %1111111111111110 %0000000000000000 1 295 ST 3D Graphics Abacus Software 4.5.1 Description of the new subroutines: menu: Display a small menu and wait for a function key to be pressed. (F10 returns to Desktop immediately) testhide: Test if H or P key pressed, branch accordingly to dohide or dopaint. dohide: Calculate visible surfaces and draw. Then check if filling is required, if not, wait for <Retum>. dopaint: Fill all surfaces of rotation body and wait for <Retum>. paint a 11: Enter all surfaces of rotation body into surf addr array, sort and fill. inpmous: Enter up to 25 points (maxpoint) with the left mouse button. These points are entered through save it into the point array of the rotation body. Enough space must be reserved in the point array by entering zeros here. For entering fewer than maxpoint points end input with the right mouse button. mouspos: Wait for mouse input, also returns after keypress. Therefore it checks to see which event occured. This GEM function must be called twice for unknown reasons in order to wait once for an input. marktype: Determines the appearance of the marker set by function polymarker. markit: Call the function polymarker to set a marker. setmode: Set input mode. save it: Stores the coordinates entered with the mouse in the point array of the definition line for the rotation body. 296 Abacus Software ST 3D Graphics saveit: switch: setf orm: coord: mousform: Stores the coordinates entered with the mouse in the point array of the definition line for the rotation body. Switches the logical page to the displayed page so that the page being drawn is the page being displayed. Otherwise the filling will not be seen and the hardcopy with <Altemate> and <Help> will not function either. Change mouse form. Draw a coordinate system. Permits the definition of a user-defined mouse form whose data follows after mousforl. This new mouse form appears after F9 is pressed and looks like a snail. You can change the data in the program according to your own taste. 297 ST 3D Graphics Abacus Software 4.6 Handling several objects All subroutines discussed up to now really allow the simultaneous display of several objects. The only changes required are limited to the construction of an object definition block for each object, as well as an exchange of the makewrld routine. Let us consider the concrete example of the house from hidel. s and the changes that would be required, to construct a world system with two houses using the existing definition. The most promising approach appears to be to copy all of the house definitions (housdatx, houslin, houspla, etc.) into the corresponding arrays of the world system several times. The point coordinate arrays housdatx etc. do not present problems. They can be simply appended to the world system. A world system containing two houses would contain 52 points. More difficult is the creation of the world line array since the line definition of the individual objects, here the two houses, always starts at point offset one; the first line of every object starts at point 1 and runs to point 2 for the houses. If the world point array is extended by another house, it becomes apparent that the first line of the second house starts at point 27 of the world point array and runs to point 28, since the first 26 points belong to the first object The necessary procedure is simple: when constructing the line array from the individual object line arrays, add the total number of points in the first object to each line definition of the second object. Analagously, with three objects the sum of the points of the first two objects is added to the line definitions of the third object during construction of the world line array. The principle of the construction of the world line array is also used during construction of the world surface array, for example the first surface definition of the second house within the world surface array: 4,27,28,28,29,29,30,30,27 Furthermore, the total number of all points, lines and surfaces must be calculated and recorded. If we start with a realistic world description, the positions of the objects in this world system can change continuously-recall the airplane and the tanker truck from Section 4.1. As a consequence of this, it is necessary to 298 Abacus Software ST 3D Graphics objects belonging to it. The recreation is limited to the coordinate arrays however, since only they change. The line and surface arrays are not affected by the position change. The line and surface world arrays are created only once at the beginning of the program. The coordinate array is created twice in every main loop pass. Now to the object definition block, which contains all the information describing the individual object. The idea was to extend the available world system by one object through addition of the definition block to the existing blocks and incrementing the "object counter." Here for clarification is an object definition block in which N is replaced with the index of the current object: objectN: objNxda: .dc.l Address of the X-coordinate array of the obj. objNyda: .dc.l Address of the Y-coordinate array of the obj. objNzda: .dc.l Address of the Z-coordinate array of the obj. objNlin: .dc.l Address of the object line array objNpla: .dc.l Address of the object surface array objmrk: .dc.w Number of points in this object objNali: .dc.w Number of lines in this object objpln: .dc.w Number of surfaces on this object objNxO: .dc.w X-position of object in world system objNyO: .dc.w Y-position of object in world system objNzO: .dc.w Z-position of object in world system objNxw: .dc.w Rotation angle of obj. about X-axis objNyw: .dc.w Rotation angle about Y-axis objNzw: .dc.w Rotation angle aboutZ-axis 299 ST 3D Graphics Abacus Software The angles and also the position in the world system relate to the "rotationally neutral” point of the current object, the origin of the object definition coordinate system. As a whole, the block consists of 38 bytes, but can easily be extended with additional information, such as scale factors, etc. If two identical objects are to be created, you write two object definition blocks this is important since the creation routine finds the next block using the distance of 38 bytes between two blocks. Since two identical objects are to be created, the addresses for the two blocks are the same and only the position of the objects and perhaps the rotation angles differ. After the definition has been completed, the total number of objects, in this case two, is placed in the variable numob j: and now the total world system can be generated with a single subroutine call. Examine the definition blocks in the following listing of multil. s, in which four identical objects are already created through concatenation of four object definition blocks. Naturally, you are not limited to the creation of identical objects. You can define a new object, such as a church, and enter its definition array address and desired position into an object block. Three houses and your church will be displayed. Description of the new subroutines in multil. s: The main loop is easily changed. Here the total number of the desired objects, four, is passed and the new subroutines new_wrld and new_mark are called. new_wrld: The one-time call to the subroutine first creates the entire world system consisting of coordinate, line and surface arrays with corresponding parameter passing of the lines created, etc. Furthermore, the world parameters are passed to the variables of the link file. This assignment was previously performed by subroutine wrldset. new_mark: Change the position of an object in the world system this subroutine recreates the total coordinate system with the aid of the modified parameters and at the same time passes the world parameters to the variables of the link file. 300 Abacus Software ST 3D Graphics new_it:, surf_lin:, surf_arr: These three subroutines are called by new_wrld and new , mark and handle the actual creation of the world system from the individual object definitions. change: Change the object parameters of the individual objects. For simplification, modification is passed to all four objects. General comments on the program: Beside being able to display multiple objects, this program offers another novelty: two successive transformations of the same object. First, the four objects are "set" into the world system with new_mark: after they have first been rotated about three axes. After all objects have been ‘’rotated" in the world system you can, through control with the keyboard, rotate the entire system consisting of the four houses around a point in the world system, or move the projection plane similar to previous programs. The four houses of the system rotate around different axes of their "rotationally neutral” points at various places in the world system. The display on the screen occurs after the removal of the hidden lines with the familiar subroutine hide it:, which is used on the complete world array so that the four houses are not created through mirroring or something similar, but the hidden surfaces of all four objects are calculated in real¬ time. The hide it algorithm of this program does not recognize covering by other visible surfaces so that a house covered by other houses will be drawn. Control keys are again the cursor, help and undo keys, as well as the / * - + keys on the keypad. The speed is quite impressive. One enhancement, besides the addition of user-defined objects, is the ability to change an object’s parameters in the subroutine change: by keyboard input, for example, and to change the position of single objects in the system. 301 ST 3D Graphics Abacus Software * multil.s 22.2.1986 * * Multiple objects, four houses * * with hidden line algorithm * * * . globl .globl .globl .globl .globl .globl .text main,xoffs,yoffs,zoffs,offx,offy,offz viewx,viewy,viewz wlinxy,mouse off,setrotdp,inp_chan,pointrot wrldx,wrldy,wrldz,gnummark,gnumline,gnumpla viewx,viewy,viewz,wplane new it,new_wrld,obj2mrk, obj2pln * The program starts here—called by link-file * main: mainl: jsr apinit * Announce program jsr grafhand * Get screen handle jsr openwork * Announce screen jsr mouse off * Switch off mouse jsr getreso * Screen resolution jsr setcocli * set Cohen-Sutherland clip. jsr clearbuf move.w #4,gnumobj * announce four objects jsr pageup jsr clwork * Screen resolution jsr setrotdp * initialize obs. ref. point. jsr pagedown * Display logical screen page jsr clwork jsr inp_chan * Input and change world parameters jsr change * Change object parameters jsr new_wrld ★ create lines and surfaces 302 Abacus Software ST 3D Graphics mainlopl: jsr pointrot * rotate around observ. ref. point jsr pers * Perspective transformation jsr hideit * calculate hidden surface jsr surfdraw ★ and draw jsr pageup ★ Display physical screenpage jsr change * change object parameters and jsr new mark ★ calculate new coordinates jsr inp chan * Input new parameters jsr clwork * erase page not displayed jsr pointrot ★ Rotate around rot. ref. point jsr pers ★ Transform new points jsr hideit * Calculate hidden surfaces jsr surfdraw * and draw them jsr pagedown * Display this logical page jsr change * Change object parameters jsr new mark ★ Calculate new point coordinates jsr inp chan * Input and change parameters jsr clwork * erase physical page jmp mainlopl ★ to main loop move.1 physbase. logbase jsr pageup * switch to normal display page rts * back to link file, and end A********************************************************************** * Create the point coordinates of the world array with the * * information from the object parameter block (objectl) * *********************************************************************** new mark: move. w #0,offx move.w >■ 4-1 4-1 0 o move. w #0,offz jsr new it move.1 #viewx,pointx move.1 #viewy,pointy move.1 #viewz,pointz move . 1 #wrldx,datx move. 1 #wrldy,daty mo ve. 1 #wrldz,datz move.1 #wlinxy,linxy 303 ST 3D Graphics Abacus Software move.w gnummark, nummark move.w gnumline,numline move.w gnumpla,numsurf rts * Change the object parameter, in this case the rotation angle * in the object parameter block, which is then taken into account * * when calculating point coordinates with rnew_mark ********************************************************************** change : changwl: move. w objlyw,dO add. w #4, dO cmp.w #360,dO bit changwl sub.w #360,dO move.w dO,objlyw move.w dO,obj2xw move.w dO,obj3zw move.w dO,obj4xw move.w dO,obj4yw move.w rts dO,obj4zw *********************************************************************** * Set all world parameters for the link file variables and * create the point, line, and surface arrays of the world system *********************************************************************** new wrld: move. w #0, dO move.w dO,offx move.w dO,offy move.w dO,offz move.w proz,zobs move.w #0,dist move.1 iscreenx,xplot move.1 #screeny,yplot move.w picturex,xO move.w picturey,yO * Location of projection plane * Address of screen array * Screen center Abacus Software ST 3D Graphics jsr new it * Pass coordinates jsr surf lin * Pass lines jsr surf arr * Pass surfaces of move. w gnummark,nummark * all objects to world system move.w gnumline,numline * Total number of corners, lines move.w gnumpla,numsurf ★ and surfaces of world system move.1 #wrldx,datx * Pass parameters of world system to move.1 #wrldy,daty * link file variables move.1 #wrldz,datz move. 1 #viewx,pointx move.1 #viewy,pointy move.1 #viewz,pointz move.1 #wlinxy,linxy rts * Subroutine for creating the world system coordinate array * •ft********************************************************************** new it: new_lopl: move.1 #0,mark_it * Pointer in wrldx,wrldy,wrldz move.w gnumobj,dO * Total number of objects ext .1 dO * as counter subq.1 #1, dO * Address of first object parameter move.1 #objectl, aO * block after AO. move.1 (aO),datx * Objectldatx, daty,datz, pass move.1 4(aO),daty * addresses of point array of move.1 8(aO),datz * first object. move.1 mark_it,d7 * Offset in point array lsl.l #1, d7 * times two bytes per entry move.1 d7,d6 add. 1 #wrldx,d7 * equals offset in world system array move.1 d7,pointx * Target of transmission move.1 d6,d7 add. 1 #wrldy,d7 move.1 d7,pointy add. 1 twrldz,d6 move.1 d6,pointz * Array of world coordinates move.w 20(aO),nummark * Number of corners in the object move.w 2 6{aO),xoffs * X-offset move.w 28(aO),yoffs * Y-offset in the world system move.w 30(aO),zoffs * Z-offset 305 ST 3D Graphics Abacus Software move.w 32(aO),xangle * Rotation angle of object around move.w 34(aO),yangle * the three coordinate axes move.w 36 (aO),zangle movem.1 d0-d7/a0-a6, -(a?) * Save registers jsr matinit * Initialize rotation matrix jsr zrotate * rotate first about the Z-axis, then jsr yrotate * around Y-axis, and finally jsr xrotate * around the X-axis (matrix). jsr rotate * rotate in world coordinate system movem.1 <a7)+,d0-d7/a0-a6 move.w 20 (aO),d7 ★ Number of corners in the object ext. 1 d7 add. 1 d7,mark_it ★ as offset in point array for add. 1 #38,aO * the next object dbra dO, new_lopl * repeat, until all objects move.1 mark_it,d7 ★ have been pased. After end in move.w d7,gnummark * mark it the total number of rts it points in the world system *********************************************************************** * Pass all lines to world system, one-time call at * * program start since nothing changes in the lines *********************************************************************** surf lin: move.w gnumobj,dO ext. 1 d° subq.1 #1, dO move.1 #objectl,aO move. 1 #0,linpntr move. w #0,mark it sflnlopl: move. 1 linpntr,d7 lsl.l #2,d7 move. 1 d7,d6 add. 1 twlinxy,d7 move.1 d7, a2 move. 1 12(aO),al move.w 22(aO),dl ext. 1 dl lsl.l #1, dl subq.1 #1, dl * Total of all objects * as counter * Address of first Object par. blk. * Pointer to line array * Pointer to point array * Line pointer times four, * one lines requires four * bytes. * Start address of line array, add * to line pointer * Address of line array of object * Number of lines in this object * Number of lines times two equals * Loop counter for word transmission 306 Abacus Software ST 3D Graphics sflnlop2: move.w (al)+ ,d7 * first point of first line add. w mark it,d7 * add the offsets of current move . w d7,(a2)+ * objects, and store in world line; dbra dl,sflnlop2 * array, until all lines of this * object move.w 20(aO),d7 * Number of corners of last object add. w d7,mark it * add to corner pointer move. w 22 (aO) ,d7 ★ Number of lines ext. 1 d7 add. 1 d7,linpntr * Total number of lines add. 1 #38,aO * Object offset, distance to next dbra dO,sflnlopl ★ object. When all objects are * completed move. 1 linpntr,d7 ★ then store total number of lines move.w d7,gnumline * in the world system and rts * back s surface array of the r*********************' move.w #0,mark_it move.1 #0,plapntr move.w #0,gnumpla move.w gnumobj,dO ext. 1 dO subq.1 #l,d0 move.1 #objectl,aO move.1 plapntr,d7 add. 1 #wplane,d7 move.1 d7,a2 move. w 24 (aO),dl ext .1 dl subq.1 #1, dl move.1 16 (aO),al move.w (al),d2 ext. 1 d2 lsl.l #1, d2 * Create the array of surfaces * Counter of surfaces * Number of objects * as loop counter * Address of first object param. blk * Pointer to surface array * World surface array * Number of surfaces on this object * as loop counter * Address of surface array of the object * Number of lines of this surface * times four (one line = four bytes) 307 ST 3D Graphics Abacus Software move.1 d2,d6 lsl.l #l,d6 * complete the mult. by 4 addq.1 #2,d6 * plus 2 bytes for number of lines subq.1 #l,d2 * counter add. 1 d6,plapntr move. w (al)+, (a2) + * Number of lines in this surface sfarlop3: move.w <al)+ , d7 * From the object surface array add. w mark_it,d7 * Add point offset of the object move.w d7, <a2) + * to world surface array dbra d2,sfarlop3 * until all lines of this surface dbra dl,sfarlop2 * until all surfaces on this object move.w 20 (aO),d7 * Number of corners add.w d7,mark it * add to point offset move.w 24 (aO),d7 add.w d7,gnumpla * add to total number add. 1 #38,aO * Object offset to next object dbra dO, sfarlopl * until all objects of the world rts * and return * Input and change parameters ******************************************************************** inp_chan: jsr inkey * Read keyboard, key code in cmp.b #'D',dO bne inpwait jsr scrdmp * make hardcopy inpwait: swap dO * DO , test if cmp.b #$4d,d0 * Cursor-right bne inpl addq.w #1,ywplus * if yes, add one to Y-angle bra inpendl * increment and continue inpl: cmp.b #$4b,dO * Cursor-left, if yes then bne inp2 * subtract one from Y-angle subq.w #1,ywplus * increment bra inpendl 308 Abacus Software ST 3D Graphics inp2: cmp .b #$50,dO * Cursor-down, if yes then bne inp3 addq. w #1,xwplus * add one to X-angle increment bra inpendl inp3 : cmp .b #$48,dO * Cursor-up bne inp3a subq.w #1,xwplus * subtract one bra inpendl inp3a: cmp .b #$61,d0 * Undo key bne inp3b subq.w #1,zwplus bra inpendl inp3b: cmp.b #$62,dO * Help key bne inp4 addq ..w #1,zwplus bra inpendl inp4 : cmp.b #$4e,dO * plus key on the keypad bne inp5 * if yes, subtract 25 from position sub. w #25,dist * projection plane (Z-coordinate) bra inpendl inp5: cmp.b #$4a,d0 * minus key on the keypad bne inp6 * add. w #25,dist * if yes, add 25 bra inpendl inp6: cmp.b #$66,dO * times key on keypad bne inp7 * if yes, then subtract 15 from the ★ rotation sub.w #15,rotdpz * point Z-coordinate bra inpendl * Make change inp7: cmp.b #$65,dO * Division key on keypad bne inp8 add. w #15,rotdpz * add 15 bra inpendl inp8: 309 ST 3D Graphics Abacus Software inplO: crop -b #$44,dO * F10 pressed ? bne inpendl addq.1 #4, a 7 * if yes, jump to bra mainend * new input inpendl: move. w hyangle,dl * Rotation angle about Y-axis add. w ywplus,dl * add increment cmp. w #360,dl * when larger than 360, then subtract bge inpend2 cmp. w #-360,dl * if smaller then 360, then ble inpend3 * add 360 bra inpend4 inpend2: sub. w #360,dl bra inpend4 inpend3: add.w #360,dl inpend4: move.w dl,hyangle move.w hxangle,dl * proceed in the same manner with add.w xwplus,dl * Rotation angle about the X-axis cmp. w #360,dl bge inpend5 cmp.w #-360,dl ble inpend6 bra inpendl inpend5: sub. w #360,dl bra inpend7 inpend6: add.w #360,dl inpend7 : move.w dl,hxangle move.w hzangle,dl add.w zwplus, dl cmp.w #360,dl bge inpend8 cmp.w #-360,dl ble inpend9 bra inpendlO inpend8 : sub. w #360,dl bra inpendlO inpend9: add.w #360,dl 310 Abacus Software ST 3D Graphics inpendlO: move.w dl,hzangle rts ****************************************************************** * Determine the current screen resolution * ****************************************************************** getreso: move. w #4,-(a7) . trap #14 addq.1 #2,a7 cmp. w #2, dO bne getrl move.w #320, picturex * Monochrome monitor move.w #200,picturey bra getrend getrl: cmp. w #l,d0 bne getr2 move.w #320,picturex * medium resolution (640*200) move.w #100,picturey bra getrend getr2: move.w #160,picturex * low resolution (320*200) move.w #100,picturey getrend: rts * Hardcopy of screen, called by inp_chan * scrdmp: move.w trap addq.1 jsr rts #20,- (a7) #14 #2,a7 clearbuf 311 ST 3D Graphics Abacus Software A********************************************************************** * Initialize the rotation reference point to [0,0,0] * *********************************************************************** setrotdp: move.w #0, dl * set the initial rotation move.w dl,rotdpx * reference point move.w dl,rotdpy move. w dl,rotdpz move.w #Q,hyangle * initial rotation angle move. w #0,hzangle move.w #0,hxangle move.w #0,ywplus move. w #0,xwplus move.w #0,zwplus rts *******************************************************************, * Rotation around the rot. ref. point around all three axes *******************************************************************> pointrot: move.w hxangle,xangle * rotate the world around move.w hyangle,yangle move.w hzangle,zangle move.w rotdpx,dO * rotation ref. point move.w rotdpy,dl move.w rotdpz,d2 move.w dO,xoffs * add for inverse transformation move.w dl,yoffs move.w d2,zoffs neg .w dO neg. w dl neg. w d2 move.w dO,offx * subtract for transformation move.w dl,offy move.w d2,offz jsr matinit * matrix initialization jsr zrotate * rotate 'matrix' aboutZ-axis jsr yrotate * rotate 'matrix' about Y-axis jsr xrotate * then rotate around X-axis jsr rotate * Multiply points with the matrix 312 Abacus Software ST 3D Graphics ********************************************************************* * Set the limits of screen window for the Cohen-Sutherland * * clip algorithm built into the draw-line algorithm * * The limits can be freely selected by the user, which makes the * * draw-line algorithm very flexible. * ********************************************************************* setcocli: move.w #0,clipxule move.w # 0,c1ipyule move.w picturex,dl lsl.w #1,dl subq.w #l,dl move.w dl,clipxlri move.w picturey,dl lsl.w #1,dl subq.w #l,dl move.w dl,clipylri rts * Entry of visible Surfaces into the vplane array * ********************************************************************* hideit: move.w numsurf,dO * Number of surfaces as counter ext.1 dO subq.l #l,dO move.1 #viewx,al * The point coordinates are stored move.l #viewy,a2 * here move.l #viewz,a3 move.l #wplane,aC * here is the information for move.l #vplane,a5 * every surface move.w #0,surfcount * counts the known visible surfaces. visible: move.w (aO),dl * start with first surface. Number of ext.l dl * points on this surface in Dl. move.w 2(a0),d2 * Offset of first point on this surface move.w 4(a0),d3 * Offset of second point move.w 8(a0),d4 * Offset of third point subq.w #l,d2 * subtract one for access to point array subq.w #1,d3 * from current point offset. 313 ST 3D Graphics Abacus Software doitl: subq.w #l,d4 lsl. w #l,d2 ★ continue to multiply with two lsl. w #l,d3 lsl. w #1, d4 * and then access current move.w (al,d3.w),dS ★ point coordinates cmp.w <al,d4.w),d6 * Comparison recognizes two points bne doitl * with matching coordinates, which can move.w (a2,d3.w),d6 * occur during construction of rotation cmp. w (a2,d4.w),d6 * bodies. When two identical points bne doitl * are found, the program move.w (a3,d4.w),d6 * selects a third point for determination cmp. w (a3,d3.w),d6 * of the two vectors. bne doitl move.w 12(aO),d4 subq.w #l,d4 lsl. w #1, d4 move.w (al,d3.w),d5 * here the two vectors which lie in the move.w d5, kx * surface plane are determined through sub. w (al,d2.w),d5 * subtraction of the coordinates from move.w d5,px * two points of the surface move.w (a2,d3.w),d5 move.w d5,ky * The direction coordinates of the sub.w U2,d2.w) ,d5 * vectors are stored in the variables move.w d5,py * qx,qy,qz and px,py,pz. move.w (a3,d3.w),d5 move.w d5, kz sub. w (a3,d2.w),d5 move.w d5, pz move.w (al,d4.w),d5 * Calculate vector Q sub. w (al,d2.w),d5 move.w (a2,d4.w) ,d6 sub. w (a2,d2.w),d6 move.w (a3,d4.w),d7 sub. w (a3,d2.w),d7 move. w d5,dl * qx move . w d6,d2 * qy move. w d7,d3 * qz 314 Abacus Software ST 3D Graphics vector * the sightl mu Is py,d3 * Calculate of the cross product mu Is pz, d2 * of the vector perpendicular sub. w d2,d3 * to the current surface move.w d3, rx mu Is pz, dl mu Is px,d7 sub. w d7, dl * the direction coordinates of the move.w dl, ry * standing vertically to the surface mu Is px, d6 * are temporarily stored in rx,ry,rz mu Is py,d5 sub. w d5,d6 move.w d6, rz move.w prox,dl * The projection center serves as the comparison sub. w kx,dl * point for the visibility of a surface. move.w proy,d2 * which is acceptable for the viewing sub. w ky,d2 * situation chosen here. One can also move.w proz,d3 * use the observation ref. point as sub. w kz, d3 * comparison point. mu Is rx, dl * Now follows the comparison of vector muls ry, d2 * R and the vector from one point of the mu Is rz,d3 * surface to the projection center add.l dl, d2 * by creation of the scalar product add.l d2,d3 * of the two vectors. bmi dosight surface is visible, otherwise continue with next surface. move.w (aO),dl * Number of lines of the surface ext.l dl lsl.l #2,dl * Number of lines times 4 = space for Lines addq.l #2,dl * plus 2 bytes for the number of lines. add.l dl,aO * add to surface array for access : dbra dO,visible * to next surface. If all surfaces bra hideend * are completed, then end. ST 3D Graphics Abacus Software dosight: move. w ext .1 (aO),dl * dl * move.1 lsl.l move. 1 addq.1 dl, d2 #1,dl * aO, a4 #2,a4 * sight3: move.w (aO)+,(a5)+ dbra dl,sight3 addq. w bra #1,surfcount sight1 hideend: rts Number of lines in this surface, multiplied by two equals the number of words to be passed Access to first line of the Surface * Pass the number of the lines * and the individual lines * the number of surfaces plus * one, and work on next one ********************************************************************* * Draw surfaces entered in vplane * ********************************************************************** surfdraw: move.1 move.1 * draw surfaces with the count xplot,a4 * of surfaces passed in surfcount yplot,a5 move. 1 move.w ext .1 subq.1 bmi surflopl: move.w ext. 1 subq.1 #vplane,a6 * Description in array at address surfcount,dO * vplane, was entered by routine hideit dO #1, dO surfend (a6)+,dl dl #l,dl * if no surface was entered in array, * then end. * Number of lines on this surface * as counter of lines to be drawn. surflop2: move.1 (a6)+,d5 * first line of this surface subq.w #l,d5 lsl.w #1,d5 move.w 0 (a4,d5.w),d2 move.w 0 (a5,d5.w),d3 swap d5 * Access to screen array, which contains * display coordinates of the * points. * extract points, pass from * the routine. 316 Abacus Software ST 3D Graphics subq. w #l,d5 lsl. w #l,d5 move. w 0 (a4, d5 . w) , a2 * second point belonging to the the line move. w 0 (a5, d5 . w) , a3 jsr drawl * draw line, until all lines of this dbra dl,surflop2 * surface have been drawn and repeat dbra dO,surflopl * until all surface have been drawn. rt s * finally return. * Display and description of the same screen page * switch: move.w #-l,-<a7) move. 1 physbase,- (a7) move.1 physbase,- (a7) move. w #5,-<a7) trap #14 add. 1 #12,a7 rts * show display page in which * drawing is being made ****************************************************************** * remove all characters present in the keyboard buffer * clearbuf: clearnd: move.w #$b,-(a7) ★ Gemdos function, character in buffer trap #1 addq.1 #2,a7 tst. w dO * if yes, get character beq clearnd ★ if no, terminate move. w #1,-(a7) ★ Gemdos function CONIN trap #1 ★ repeat until all characters have addq.1 #2,a7 ★ been removed from the buffer bra clearbuf rts . even 317 ST 3D Graphics Abacus Software • data housdatx: .dc. w -30,30,30,-30,30,-30,-30,30,0,0,-10,-10,10,10 .dc.w 30,30,30,30,30,30,30,30,30,30,30,30 housdaty: .dc. w 30,30,-30,-30,30,30,-30,-30,70,70,-30,0,0,-30 .dc.w 20,20,0,0,20,20,0,0 .dc.w -10,-10,-30,-30 housdatz: .dc.w 60, 60, 60, 60,-60,-60,-60,-60, 60,-60,60,60, 60, 60 • dc.w 40,10,10,40,-10,-40,-40,-10 .dc.w 0,-20,-20,0 houslin: .dc.w 1,2,2,3, 3, 4,4,1,2,5,5, 8, 8,3, 8, 7,7,6, 6, 5, 6,1,7,4 .dc.w 9,10,1,9,9,2,5,10,6,10,11,12,12,13,13,14 .dc.w 15, 16, 16, 17, 17,18,18, 15, 19,20,20,21,21,22,22,19 .dc.w 23,24,24,25,25,26,26,23 318 Abacus Software ST 3D Graphics * here is the definition of the surfaces belonging to the house * houspla: .dc.w 4,1, 2,2,3,3, 4,4,1, 4,2,5, 5,8, 8,3, 3,2 .dc.w 4,5,6,6,7,7,8,8,5,4,7,6,6,1,1,4,4,7 .dc.w 4,4,3,3,8,8,7,7,4,4,2,9,9,10,10,5,5,2 .dc.w 4,10,9,9,1,1,6,6,10,3,1,9,9,2,2,1 .dc.w 3,5,10,10,6,6,5,4,11,12,12,13,13,14,14,11 .dc.w 4,15,16,16,17,17,18,18,15,4,19,20,20,21,21,22,22,19 .dc.w 4,23,24,24,25,25,26,26,23 hnummark: .dc.w 26 * Number of corner ■ points of the house hnumline: .dc.w 32 ■* Number of lines of the house hnumpla: .dc.w 13 ★ Number of surfaces of the house hxangle: .dc.w 0 * Rotation angle of house about X-axis hyangle: .dc.w 0 * H II II Y-axis hzangle: .dc.w 0 * 11 II M Z-axis xwplus: .dc.w 0 * Angle increment about X-axis ywplus: • dc.w 0 * Angle increment about Y-axis zwplus: .dc.w 0 * Angle increment about Z-axis picturex: .dc.w 0 * Definition of zero point on the screen picturey: .dc.w 0 ★ entered by getreso rotdpx: .dc.w 0 rotdpy: .dc.w 0 rotdpz: . dc. w 0 rlzl: .dc.w 0 normz: .dc.w 1500 .bss 319 ST 3D Graphics Abacus Software plusrot: .ds. 1 1 first: . ds. w 1 second: .ds. w 1 deltal: .ds. w 1 worldpla: .ds. 1 1 .data plag: .dc .b 1 . even .bss diffz: • ds .w 1 dx: .ds .w 1 dy: .ds .w 1 dz: .ds. w 1 wrldx: .ds. w 1600 * world coordinate array wrldy: • ds .w 1600 wrldz: .ds .w 1600 viewx: • ds. w 1600 * view coordinate array viewy: .ds .w 1600 viewz: • ds. w 1600 screenx: .ds. w 1600 * screen soordinate array screeny: .ds. w 1600 wlinxy: .ds. 1 3200 * line array wplane: .ds. 1 6600 * surface array vplane: .ds. 1 6600 * surface array 320 Abacus Software ST 3D Graphics space: . ds. 1 2 pladress: .ds. 1 3000 * surfcount: . ds. w 1 numsurf: . ds. w 1 zcount: . ds. 1 1 * zsurf: . ds . w 1 * ***************************** .data gnumobj: .dc.w 2 gnummark: . dc. w 0 gnumline: .dc.w 0 gnumpla: .dc.w 0 mark_it: .dc. 1 0 linpntr: .dc. 1 0 plapntr: .dc. 1 0 objectl: objlxda: .dc. 1 housdatx objlyda: .dc. 1 housdaty objlzda: .dc. 1 housdatz objllin: .dc. 1 houslin objlpla: .dc. 1 houspla objlmrk: .dc.w 26 objlali: .dc.w 32 objlpln: .dc.w 13 objlxO: .dc.w 150 objlyO: .dc.w 100 objIzO: .dc.w 0 objlxw: .dc.w 20 objlyw: .dc.w 0 objlzw: .dc.w 0 object2: obj2xda: • dc. 1 housdatx obj2yda: .dc. 1 housdaty obj2zda: .dc. 1 housdatz * surface array 321 ST 3D Graphics Abacus Software obj21in: .dc.1 houslin obj2pla: .dc.l houspla obj2mrk: .dc.w 26 obj2ali: .dc.w 32 obj2pln: .dc.w 13 obj2x0: .dc.w -150 obj2y0: .dc.w 100 obj2z0: .dc.w 0 obj2xw: .dc.w 0 obj2yw: .dc.w 20 obj2zw: .dc.w 0 object3: obj3xda: .dc.l housdatx obj3yda: .dc.l housdaty obj3zda: .dc.l housdatz obj31in: .dc.l houslin obj3pla: .dc.l houspla obj3mrk: .dc.w 26 obj3ali: .dc.w 32 obj3pln: .dc.w 13 obj3x0: .dc.w -150 obj3y0: .dc.w -100 obj 3z0: • dc.w 0 obj3xw: .dc.w 0 obj3yw: .dc.w 20 obj3zw: .dc.w 0 object4 : obj4xda: .dc.l housdatx obj4yda: .dc.l housdaty obj4zda: .dc.l housdatz obj41in: .dc.l houslin obj4pla: .dc.l houspla obj4mrk: .dc.w 26 obj4ali: .dc.w 32 obj4pln: .dc.w 13 obj 4x0: .dc.w 150 obj4y0: .dc.w -100 obj4z0: .dc.w 0 322 Abacus Software ST 3D Graphics obj 4xw: .dc. w 0 obj 4yw: .dc.w 0 obj4zw: .dc. w 0 .bss sx: .ds.w 1 sy: .ds.w 1 sz: .ds.w 1 px: .ds.w 1 py: .ds.w 1 pz: .ds.w 1 rx: .ds.w 1 ry: .ds.w 1 rz: -ds.w 1 qx: .ds.w 1 qy: .ds.w 1 qz: .ds.w 1 kx: .ds.w 1 ky: .ds.w 1 kz: • ds.w 1 ********************** .data • even maxpoint: .dc. 1 25 mousx: .dc.w 0 mousy: .dc. w 0 mousbut: .dc. w 0 kybdstat: .dc.w 0 altx: .dc.w 0 alty: .dc.w 0 newx: .dc.w 0 newy: .dc.w 0 323 ST 3D Graphics Abacus Software addrssx: .dc. 1 1 .data prox: .dc.w 0 * Coordinates of Projection proy: . dc. w 0 * Center on the positive proz: • dc.w 1500 * Z-axis • data of fx: .dc. w 0 * transformation during Rotation of fy: . dc. w 0 * to Point [offx,offy,offz] of f z: .dc.w 0 xoffs: .dc. w 0 * Inverse transformation to point yoffs: .dc.w 0 * [xoff,yoffs,zoffs] zoffs: .dc.w 0 .bss loopc: .ds. 1 1 .end Desk File View Options A:\ D:\ F:\3DU0RK.DIRS 253882 butes used 1 1442236 butes used In 129 itens, ■ 333956 bytes used In X PRINTERS 8 TUTORIAL C COMV NLlfl OUTPUT SPLIT STANDARD TEXTPRO TUTORIAL XTTUTORI FRY TTP PRO PRG TTP PRT PRG TXT TOC BASIC BASIC BASIC BASIC1 PRG RSC MRK BAR 138344 11-21 4648 11-21 346 11-26 14801 11-26 OPEN APPLICATION Nane: BATCH .TTP Paraneters; . asllnk grllnkl miltll)- I OK 1 I Cancel HOUSE1 PRG HOUSE1 S MAIK1 PRG KAIN1 S HAIN1C0 PRG MAIN1C0 S MEKU1 PRG HENU1 MULTI1 MULTI1 PAINT1 PRINT1 ROTATE1 PRG R0TATE1 S S PRG S PRG S 1 0 a? a? min \trrm ■n'v.-r rrrwt j-amattfil jS33HEflB33i ■BSffiSMI 324 Abacus Software ST 3D Graphics 5. Suggestions for additional development One application of this program module for manipulating three- dimensional objects that will occur to almost everyone is a flight simulator. The last program can in fact be used as a basis for a flight simulator. We are missing the description of the position of the airplane in the world system as well as a modified point rot routine. The modified point rot routine, after rotation around the reference point, should not transform all of the world coordinates back to the old coordinate origin, which occurred in the old point rot routine by adding the reference point coordinates after the rotation. Furthermore, houses do not change position in the world system of a flight simulator and for an airport other structures must be developed (hangar, tower). In addition, fields, forests, and landing strips can be simulated with simple rectangular surfaces. The position of the airplane, or to be exact, the center of its cockpit windshield, in the the reference point in the world system for all transformations to follow, especially that of the creation of the view system. For simulation of airplane movement, the reference point must be manipulated with keyboard input. This input must affect both the point coordinates as well as the orientation of the plane in space. The orientation of the airplane in space- is described with the three angles (hxangle,hyangle,hzangle) so that even adventurous flight situations (spins) can be simulated. For adjustment of the world system to the airplane system the following operations are required: 1. Move the coordinate origin of the world system to the cockpit center by subtracting the cockpit windshield center-coordinates from all point coordinates. 2. Rotate of the displaced world system about the three rotation angles which describe the position of the airplane in relation to its three axes. 327 ST 3D Graphics Abacus Software 3. Remove hidden surfaces with hideit, noting that the reference point for the calculation of the vector S through point [0,0,0], the cockpit center-point (coordinate origin of the view system) is chosen and not the projection center, which of course can also be freely selected in this observation model. From the endpoint of vector S the direct result: all objects outside the cockpit window are, if they satisfy the criteria for visibility, visible. 4. Projection on the screen through the perspective transform routine. After the observed world is displayed, the parameters such as the position of the airplane in the world system or the position of other objects in the world system, such as a second airplane, can be changed. Now the procedure described above is called again and this cycle repeated continually. 328 Abacus Software ST 3D Graphics 5.1 Light and Shadow To enhance the program module to correctly define a light source, as in section 2.8, it is necessary to have the vectors L, i.e. the vector, which points from each surface to the observation reference point as well as the vector N, which points from the light source to the current surface, as unit vectors of length one. One should divide the vector coo rdinates (x, y, z) by the root of the sum of its squares V(x2+y2 + z 2). Furthermore, the data structure of the objects must be changed since you want to shade the surfaces according to the light intensity and not according to their Z- coordinates. It is possible to enter the intensity of every surface in the extended surfaddr array and give each surface an individual shading pattern, either through comparison of the light source vectors or completely at random. 329 ST 3D Graphics Abacus Software 5.2 Animated Cartoons In principle even this application has already been realized in program multi. s. You simply create more objects in a world system and then changes their position and place in the system continuously. The world line and surface arrays, as we have seen, need be created only once while the coordinate array is created with every pass through the main loop. After the line surface array has been constructed, you have free choice in the number of displayed objects, i.e. you can define, for example, ten objects through object definition blocks but at the creation of the comers you could only actually create and display. One possible application is moving text where the letters are three dimensional objects. You could have several letters "fly" together from various directions and assemble them on the screen into a word. The complete word could then be rotated around some point. The individual letters could even be constructed with the mouse. 330 Abacus Software ST 3D Graphics Appendix A: Number systems Every number, in any number system, is represented by a sequence of digits. This sequence may be interrupted by a decimal point. We can write the following for the digit sequence: (...a4a3a2aiao.a-ia-2a-3a-4.")b = + a4*b4 + a3*b3 + a2*b2 + ai*bl + ao*bO + a-i*b _ l + ... Here the coefficients a-4 to a4 represent the individual digits of the number and b is the number base. Here is an example of the most commonly used number system, the decimal system: (3423.87)10 = 3*103 + 4*102 + 2*101 3*100 + 8*10-1 + 7*10-2 = 3000 + 400 + 20 + 3 + 0.8 + 0.07 = 3423.87 Two number systems often encountered, in computer programming, are the binary (base 2) and the hexadecimal systems (base 16). Binary uses only the two numbers 0 and 1 as digits. An example: 1110010010010= 1*212+1*211+1*210+1*27+1*24+1*21= 4096 + 2048 + 1024 + 128 + 16 + 2 = 7314 Numbers in the hexadecimal system with base 16 are generally indicated by a leading dollar sign ($). For representation of numbers in this format, the standard ten digits from 0 to 9 are not enough. For this reason the first six characters of the alphabet are added (A through F). A has the value of 10, and F means 15. It is especially easy to convert between binary and hexadecimal. Four binary digits (4 bits) are grouped together, starting from the decimal point, to form one hexadecimal digit. 333 ST 3D Graphics Abacus Software The unwieldy binary number 1110010010010 becomes the hexadecimal number $1C92. The conversion into the Decimal system is done in the same manner as for the binary system. $1C92 means therefore: 1*163 + 12*162 + 9*161 + 2*160 = 1*4096 + 12*256 + 9*16 + 2*1 = 7314 334 Abacus Software ST 3D Graphics Appendix B: Analytical geometry of planes and space The cartesian coordinate system is defined as a system of perpendicular lines in which the horizontal line is designated as the X-axis (abscissa) and the line perpendicular to it is called the Y-axis (ordinate). The intersection of the two lines is the origin of the system. Now all points within the system can be defined unambiguously by specifying their coordinate values (x,y). A line in such a system is defined by two points which belong on the line. All points on the line can be ascertained with the following equation. y-yi x-xl y2 : yl x2-xl for (x2-xl) <> 0 In this two point format, the expression (y2-yl)/(x2-xl) gives the slope m of the straight line, which simultaneously represents the tangent of the angle between the line and the X-axis (phi). 335 ST 3D Graphics Abacus Software y2-yi x2-xt Figure B.l: Line in the plane With the definition of the slope m as well as the axis intersection a, the intersection of the line with the Y-axis, we get what is called the normal form of the straight line equation. y = m*x + a With this equation you can calculate all points on the line by introducing various X values into the above equation, knowing the slope m and axis intersection a. For the middle-point of a straight line which connects two points (PI, P2), we can easily calculate the coordinates of this segment: Xm = xl+x2 "T* Ym = yl+y2 2 336 Abacus Software ST 3D Graphics The two equations above are used in the Cohen-Sutherland clipping algorithm. The geometry of a plane is just a special case of the geometry of space and therefore the same laws apply to a straight line in space as to a straight line in a plane, i.e. two points are also sufficient to define a point in space. One difference from the plane is the Z-axis which, if one leaves the X and Y axis unchanged, can point in different directions. Depending on the direction used, this system is called a right-hand or left-hand system. They differ therefore only in the orientation of the Z-axis. Figure B.2 An easy way to distinguish between the right- and left-hand systems as well as all operations within the system is possible with the aid of a screw (imagine simply a normal screw inside the system). The screw transfers a rotating motion into a movement along the rotation axis and there are basically two types of screws: those with left-handed threads and those with right-handed threads. For a complete system description, we still need to know how positive angles are measured and for equalization of both coordinate systems the following definition is agreed upon: 337 ST 3D Graphics Abacus Software Rotation about the: positive angle is measured: Z-axis from +X to +Y axis Y-axis from +Z to +X axis X-axis from +Y to +Z axis With the aid of this definition we can say for the system and the screw: If a screw is placed in such a system (in the direction of a coordinate axis) and the screw is turned about a positive angle (see above definition), then the screw moves in the direction of a positive coordinate axis. You can determine the position axis of a coordinate system through the definition of the positive angle as well as the selection of the screw, or you can recognize the type of an existing coordinate system. As an explanation, in a right-hand system the right-handed screw moves in the direction of a positive coordinate axis when rotated about a positive angle. On the other hand, a left handed screw in a left-hand system rotated about a positive angle will also move in the direction of positive coordinate axis. Since in our country, screws with right-handed threads are most common, we shall follow the positive rotations of a right-handed screw in a right-hand system. 338 Abacus Software ST 3D Graphics Figure B.3: Screws in a right-hand system 339 ST 3D Graphics Abacus Software Two points in space or in a plane are sufficient to describe a line. Under consideration of Z-coordinates the following relationships hold: y-yl = y2-y 1 z-zl _ z2-zl x-xl x2-xl x-xl x2-xl Using a parameter u, which can assume real values between -infinity and +infinity, all points on a line running through points Pl[xl,yl,zl] and P2 [x2, y2, z2 ] can be determined. For individual coordinates the values are: x = (x2-xl) * u + xl y = (y2-yi) * u + yl z = (z2-zl) * u + zl If we use only u real numbers between 0 and 1, all points on the line between PI and P2 can be calculated. The line would not run beyond PI and P2, but would be cut off at the two points. From the lines we get a vector, which has a definite direction in space. In our example it points from PI to P2. A vector is a directional line, the connecting line between two points in a coordinate system. The coordinates of the vector are calculated by subtracting the point coordinates. The vector is therefore indicated by the vector coordinates and its direction. The direction is shown in the illustration by an arrow. A vector can be moved along its axis without consequences for the total system, since only the length and direction are of significance. 340 Abacus Software ST 3D Graphics The vector S in Figure 6.3.4 is given by its vector coordinates S [sx, sy, sz] = [x2-xl,y2-yl,z2-zl] and its value, the length of the distance S, can be determined as follows: Value S = |S| = V (sx 2 + sy2 + sz2) A unit vector is a vector whose value is one. If you want to generate a unit vector to a given vector S, a vector which points in the same direction as S but has a value of one, the vector coordinates of the unit vector are I [ix,iy,iz]: sx sy . sz ix =—— iv — iz =rr:— Dividing the individual vector coordinates of vector S [sx,sy,sz]by the length of vector S results in the vector coordinates of the unit vector. 341 ST 3D Graphics Abacus Software Various operations can be performed on the vectors and those important for our purposes are: 1 . The scalar product (A-B) 2. The cross product (AXB) B.l Scalar Product The scalar product is the sum of the products of the individual vector coordinates and is important to determine angles (phi) between two vectors (A,B). A-B = ax*bx+ay*by+az*bz = | A | *| B |*cos(phi) A-B =V ( (ax 2 +ay 2 +az 2 ) * (bx 2 +by 2 +bz2) ) *cos (phi) See also Figure 2.7.5. B.2 Cross Product The cross product (AxB), in contrast to the scalar product, is not a real number but another vector (C). The resultant vector stands perpendicular to the plane between the vectors A and B and together with them forms a new coordinate system. The rule of the screw helps us again in the determining the direction of the resulting vector: In a right-hand system the result vector (C = [cx,cy,cz]) of the cross product points in the same direction in which a screw with right- handed threads would move from A to B when turned. The vectors A, B, and C form a right-hand system. Similarly for a left-hand system: if one turns a left-threaded screw from A to B , then C points in the direction in which^tfie~stxew would move. This connection can be seen easily in Figure 6.3.5 afad in our program is responsible for the recognition of visible, surfaces) Abacus Software ST 3D Graphics Figure B.5 To determine the result vector C [cx,cy,cz] one proceeds as follows: A*B = [ ax*bz-az*by, az*bx-ax*bz, ax*by-ay*bx] 343 ST 3D Graphics Abacus Software Appendix C: Matrix calculations A matrix (m,n) is a square number system consisting of m by n numbers. an an an ... am a2i a22 a23 ... a 2 n a3i a32 a33 ... 33n ami 4m2 Sm3 ... amn aik where i = 1,2.. .m and k = ---- -HIV A ulllVj Uliu U1V VIVUlUiUJ aik,a 2 ic,.-amk form the kth column of the matrix. If the number of columns is equal to the number of rows (m=n), A is called a square matrix. A few rules can be stated for matrix calculation. 1. Matrices are designated with uppercase letters (A-Z). The individual elements of a matrix carry the corresponding lower case letter (a-z). 2. The element aik is located in the ith row, kth column of matrix A. i is the row index and k is the column index. 3. The matrix A(m,n) is of the type (m,n) and is defined as a two-dimensional matrix with m rows and n columns. 4. Matrices with one row and any number of columns, of the type (l,n), are called row vectors and those of type (n,l) are called column vectors. 344 Abacus Software ST 3D Graphics C.l Adding matrices The addition of matrices is defined only for matrices of the same dimensions. Here is an example with two (3,3) matrices, A with the elements aik and the matrix B with the elements bik- During addition, the sum matrix S is created with elements sik- S=A+B. 1 2 3 A= 4 5 6 7 8 9 2 C = A+B = 8 14 1 2 3 B = 4 5 6 7 8 9 4 9 10 12 16 18 The elements of the sum matrix result from: sik = aik + bik for i,k from 1 to 3. The limits of the variables i and k are written in mathematical form: i,k = 1(1)3. The value in front of the parentheses is the start value, the value in the parenthesis is the increment and the last number designates the final value of the variables. In this example, i and k take values of one through three with an increment of one. These are the numbers 1,2,3. During matrix addition, one adds the elements which are in the same place in each matrix, to obtain the elements of the sum matrix S. One proceeds in the same manner when multiplying of matrix A with a constant factor f ac. The elements of the product matrix P are calculated by multiplying each element in A by the factor. pik = fac * aik i,k= 1(1)3 345 ST 3D Graphics Abacus Software C.2 Multiplying Matrices The multiplication of two matrices A and B is somewhat more complex than addition and has some limitations. The product of two matrices is only defined when the number of columns of A matches the number of rows in B. For two square matrices with i=k=constant, the multiplication is always defined. The product of two matrices A (ajj) and B (bjk) is defined as follows: A is a matrix of type (m,l) and B is of type (l,n), then the product of the matrices A and B is A*B, the result matrix P is (pik), whose elements are calculated in the following manner: pik = sum of j=l to 1 over aij*bjk with i - l(l)m and j = l(l)n. This connection can be recognized in the following example. A= 1 2 B= 5 6 3 4 7 8 C = A*B = 1*5 + 2*7 1*6+ 2*8 3*5 + 4*7 3*6 + 4*8 19 22 43 50 The result matrix P therefore contains the same number of lines as the multiplicand A and the same number of rows as the multiplier B. In regard to matrix multiplication there is a neutral element, i.e. for every matrix A there is a matrix N with which A can be multiplied without changing the original matrix. A*N=A. N is called the unit matrix and the elements of the diagonal are one. All others have the value zero. Moreover, the associative and the distributive law are valid during multiplication. A*(B*C) = (A*B)*C Associative Law A*(B+C) = (A*B)+(A*C) Distributive Law 346 Abacus Software ST 3D Graphics The commutative law does not hold for matrix multiplication. This means A*B is not necessarily equal to B*A. The order of the multiplication is not arbitrary, as you see, and must be observed. 347 ST 3D Graphics Abacus Software Appendix D: Bibliography [1] Foley James D., van Dam A., Fundamentals of Interactive Computer graphics, Addison Wesley Publishing Company (1984) [2] Harrington Steven, Computer Graphics, McGraw-Hill (1983) [3] Newman William M., Principles of Interactive Computer Graphics, McGraw-Hill (1984) [4] Knuth Donald E., The Art of Computer Programming Volume 2 Seminumerical Algorithms, Addison-Wesley Publishing Company (1981) [5] Kane Gerry, Hawkins Doug, Leventhal Lance, 68000 Assembly Language Programming, McGraw-Hill (1981) [6] Bruckmann, Englisch, Gerrits, Atari ST Internals, Abacus Software (1986) [7] Szczepanowski, Gunther, Atari ST GEM Programmer’s Reference, Abacus Software (1986) 348 Abacus Software ST 3D Graphics INDEX 3-D wire models 4 3-D wire models 56 68000 computer 78 A Abscissa 335 Apple Macintosh 4 Assembler 88, 89 Associative 346 Axis-symmetrical objects 169 B BASIC1.S 92 Batch file 90 Binary system. BIOS 88 C C programming language 4 CAD systems 3 CAD-CAM 3 Cartesian coordinate system 7, 108, 249, 335 CAT scans 3 Clip algorithm 122 Cohen-Sutherland clipping algorithm 122, 337 Color monitor 219 Column vectors 344 Commodore Amiga 4 Computer animation 3, 56 Computer science 122 Convex polygons 57 Coordinate arrays 299 Coordinate origin 11 Cray II 3,4,56 Cross product 214, 342 D Data system 117 Decimal system 333, 334 Definition block 298-300 Definition line 169, 189 Desktop 296 Digital Research 88 Direction Binary number 334 Display coordinate system 7 Display of reveral objects 298 Distributive law 346 Draw-line-algorithm 107 E Extended coordinate system 31 Extended-BIOS 88 G GEM functions 88 GEM-DOS 88 Global variables 152 GRLINK1.S 123 349 ST 3D Graphics Abacus Software H Harddisk 4 Hexadecimal number 334 Hexadecimal system 333 Hidden lines 4, 248, 249, 301 Hidden surfaces 217, 301 HIDE1.S 198 HOUSE1.S 155 I Indirect illumination 69 Inverse transformation matrices 41 L Left coordinate system 30, 34,47,48, 59, 64, 337, 338, 342 Line array 298 Link file 170 Linking programs 89 M-N Machine language 4, 89 Matrices 344 Matrix addition 345 Matrix multiplication 119,346 MENUl.S 257 Metacombco Editor 107 Monochrome monitors 219 Motorola MC68000 3-4 MULTII.S 302 Normal vector 116 O Object definition block 299-300 Object definition coordinate system 300 Observation coordinate system 44 Observation direction point (ODP) 43, 44,116,216 Observation reference point (ORP) 43, 44,46 Observation window 43 Observer system 118 Operating system 30, 34,107,218, 248 Ordinate 335 P PAINT1.S 221 Pascal 4 Perspective transformation 31,119, 122 Picture coordinate system 8-9 Plot-point routine 80 Point coordinate arrays 298 Point light sources 70 Polygon 11 Polymarker 296 Projection 38,51 Projection center 50,116,167, 214 Projection plane 50, 51, 56 R Real time 3-D graphics 191 Reflection coefficient 69,71 RELMOD 88 350 Abacus Software ST 3D Graphics Relocator 89 Right coordinate systems 64,122,214, 337, 338, 342 ROTATEl.S 172 Rotating definition line 170 Rotation 16, 32, 35, 37,41 Rotation body 170,188,296 Rotation line 170,249 Rotation matrices 42,191 Rotation reference point 167 Row vectors 344 S Scalar product 214, 342 Scaling 10, 12 Scaling matrix 32 Scan line algorithm 56 Screen memory 79 Screen pages 87 Screen switch routine 122 Screen system 118 Shading routine 218,248 Shading surfaces 248 Sine table 78 Size manipulation 120 Slope 81,335,336 Square matrix 344 Straight line equation. 336 Structogram 246 Surface normal vector 216 Surface world arrays 299 T Unit vector 341 USA Today 3 User-defined objects 301 vectors 38, 340 View coordinate system 8,11,15,43, 116,118,119, 122,214 Visible surfaces 296 W Window size 165 Wire model mode 249 World array 298, 301 World coordinate system 8, 11,15,44, 109,116,117,119, 120,121,122,165, 188,217,246,298, 299, 300, 301 World parameters 300 World surface array 298 Taylor series 76 Transformations 31,121 Unit matrix 346 351 Optional Diskette For your convenience, the program listings contained in this book are available on an SF354 formatted floppy disk. You should order the diskette if you want to use the programs, but don’t want to type them in from the listings in the book. All programs on the diskette have been fully tested. You can change the programs for your particular needs. The diskette is available for $14.95 plus $2.00 ($5.00 foreign) for postage and handling. When ordering, please give your name and shipping address. Enclose a check, money order or credit card information. Mail your order to: Abacus Software P.O. Box 7219 Grand Rapids, MI 49510 Or for fast service, call 1- 616 / 241-5510. AA Rated Software Atari and Abacus A simple-to-use and versatile database manager. Features help screens; lightning-fast operation; tailorable display using multiple fonts; user-definable edit masks; capacity up to 64,000 records. Supports multiple files. RAM-disk support for 1040ST. Complete search, sort and file subsetting. Interfaces to TextPro. Easy printer control. $49.95 ST TextPro Wordprocessor with professional features and easy-to-use! Full-screen editing with mouse or keyboard shortcuts. High speed input, scrolling and editing; sideways printing; multi-column output; flexible printer installation; automatic index and table of contents; up to 180 chars/line; 30 definable function keys; metafile output; much more. $49.95 PaintPro ^ /*V ^ C. Create double¬ sized pictures Fohrq ^- PaintPr > Multiple J windows For creative illustrations on the ST r ST PaintPro A GEM™ among ST drawing programs. Very friendly, but very powerful design and painting program, A must for everyone's artistic or graphics needs. Use up to three windows. You can even cut S paste between windows. Free-form sketching; lines, circles, ellipses, boxes, text, fill, copy, move, zoom, spray, paint, erase, undo, help. Double-sized picture format. $49.95 ST Forth/MT Powerful, multi-tasking Forth for the ST. A complete, 32-bit implementation based on Forth-83 standard. Develop¬ ment aids: full screen editor, monitor, macro assembler. 1500+ word library. TOS/LINEA commands. Floating point and complex arithmetic. $49.95 AssemPro The complete 68000 assembler development package for the ST PCBoard Designer Create printed circuit board layouts ^ wuMmfimmWiiWilsmm Auto routing, component list, pinout list, net list ST AssemPro PCBoard Designer Professional developer's package Interactive, computer aided design includes editor, two-pass interactive package that automates layout of printed assembler wrth error locator, online help circuit boards. Auto-routing, 45° or including instruction address mode and 90° traces; two-sided boards; pin-to-pin, GEM parameter information, pin-to-BUS or BUS-to-BUS. Rubber- monitor-debugger, disassembler and banding of components during place- 68020 simulator, more. $59.95 ment. Outputs pinout, component and net list. $395.00 Call now for the name of the dealer nearest you. Or order directly using your MC, Visa or Amex card. Add $4.00 per order for shipping. Foreign orders add $10.00 per item. Call ( 616 ) 241-5510 or write for your free catalog. 30-day money back software guarantee. Dealers inquires welcome-over 1400 dealers nationwide. ST and 1040ST are trademarks ot Atari Corp. GEM is a trademark at Digital Research Inc. Abacus mi P.0. Box 7219 Dept.NB Grand Rapids, Ml 49510 Phone 616/241 -5510 • Telex 709-101 • Fax 616/241-5021 PRESENTING THE ST Gives you an indnpth look at thla • •nietlonal n.w computsr. Discuss as tha aichltactura ot tha ST. work¬ ing with GEM. tha mouaa. operating system, all tha varioua intarfaoaa, tha 68000 chip and Ita inetructlons. LOGO 290 pp *16 86 ST Beginner's Oulda Writtan tor tha Srtthand ST user Gat a baalc undaratand- ing ol your ST Explora LOGO and BASIC from tha ground up Simpla explan atlona o' tha hardware and Intarnal workings o' tha ST llluatratona. diagrams Gloaa- ary indax. 200pp $i« 95 ST INTERNALS Eaaantlal guida to tha Insida Inlormation ol tha ST Datailad daacrlptlona ol ■ound and graphica chips. Intarnal hardware, I/O porta, ualng GEM Commented BIOS Hating. An Indiapan aibla ralaranea lor your ST library «0pp »'«» OEM ProgrammeCe Rat. For aarioua programmara naadlng detailed nlctmation on GEM Praaantad in an oony-to-undoratand format. All aiamplaa ara In C and aaaamBty language Covara VOI and AES function# No aarioua programar ahould ba without dIOpp SI 9 99 MACHINE LANGUAGE Program In tha lastaat lang¬ uaga lor your ATARI ST Laarn 68000 Meernbly lang. uaga. ita numbanng ayatam, uaa cd raglatara. atructura S Important datarla of Inatruc- tion aat, and uaa ol intarnal ayatam routines Gaarad lot tha ST 280pp *19 95 Fantaatlc collection ol pro- grama and Into lor tha ST Complata programa Induda. euperlast RAM dtak; lima- aadng prlntar apoolar, color print hardcopy plotter output hardcopy, craating access orlaa Monay aavlng tricks andtipa 2S0cc *19 99 ST GRAPHICS 4SOUNO Datailad guida to graphica and aound on tha ST. 20 6 3D kjnctlon ptottara. Molr* patlarna, graphic memory and varioua reaolutiona. (ractala, recursion waveform generation Eianpiet written In C, LOGO. BASIC and ModMe2 250pp 519 99 ST LOGO OUIOE Taka contd of your ST by learning ST LOGO—tha aaay to uaa. powerful languaga Topic* Induda 61a handling, recurslon-HHbert A Sierpinskl curvaa. 2D and 30 (unction plots, data atructura. error handling Helpful guide lor ST LOGO users. *19.99 ST PEEKS * POKES Enhance your programa with tha aiamplaa found within thla booh Explorea ualng different languages BASIC. C. LOGO and machine languaga. using various intarlacaa. memory uaaga, reading and saving horn and todlak, more 280pp $16 95 BASIC Training Guide Thorough guida lor Isarning ST BASIC programming. Detaled programming funda¬ mentals, commands descilp- tiona. ST graphics 8 aound. using GEM in BASIC, lila management, disk operation. Tutorial problems give hand* on experience. 300pp $16 95 BASIC to C Move up from BASIC to C. If you're elreedy a BASIC programmer, you can laarn C all that much Merer Parallel examples demoslrate tha programing techniques and constucts In both languages. Variables pointers, arrays, daU atructura 290pp*19 99 30 GRAPHICS FANTASTICI Rotate, zoom, and shade 3D objects. All programa written In machine language (or high speed. Learn the mathematics behind 3D graphica. Hidden line removal, shading With 3D pattern maker and animator. $2* 96 The ATARI logo and ATARI ST era Swdamerkaof Alan Carp. Abacusiuiiiiiiil Software P.0. Box 7219 Dept. A9 Grand Rapids, Ml 49510 • Telex 709-101 ■ Phone (616) 241-5510 Optional diskettes are available tor all book titles at $14.95 Call now for the name of your nearest dealer. Or order directly from ABACUS with your MasterCard, VISA, or Amex card. Add $4.00 per order for postage and handling. Foreign add $10.00 per book. Other software and books coming soon. Call or write for your free catalog. Dealer inquiries welcome-over 1400 dealers nationwide. M n V \ 1 Q> $&></■> O ° < D o / O 3 L> ^ 1 1 T C> / 150 j 0 / D ) l 0 1 1 0 1 o n\ $ j C?D O °1 h I OO) tp \ o / -o ( 1 c l J II 'X & ! ! ?£> IX X3 1 / of iV £ j l t V /f f in/ FANTASTIC! Learn how to create impressive and fascinating 3D graphics on the Atari ST. Covers introductory concepts and background materials, graphic animation, using the assembler and much more. Learn real-time animation with dozens of graphic routines. 3D Graphics is an amazing book for all programmers interested in advanced level graphics. Some of the areas covered include: • 2D & 3D Transformations • Spatial projection • Hidden lines & surfaces • Screen manipulation • Data structure for 3D objects • Rotation of objects • Object animation • Light and shadows About the author: Uwe Braun is a member of the Data Becker Product Development team, based in Duesseldorf, W. Germany. He is a knowledgeable machine language programmer with extensive graphics programming experience. isbn o-nibMai-bn-D The ATARI logo and ATARI ST are trademarks of Atari Corp. A Data Becker book published by a I You Can 0n ImiHHiixil e + Abacus \mm Software P.0. Box 7219 Grand Rapids. Ml 49510 - Telex 709-101 - Phone 616/241-5510