Table Of ContentFast Aigorithms for 3D-Graphics
Georg Glaeser
Fast Aigorithms for
3D-Graphics
With 94 Illustrations
With a diskette
Springer Science+Business Media, LLC
Georg Glaeser
Lehrkanzel für Geometrie
Hochschule für Angewandte Kunst
Oskar Kokoschka Platz 2
A-IOIO Wien
Austria
Library of Congress Cataloging-in-Publication Data
Glaeser, Georg.
Fast algorithms for 3D-graphics : with 94 figures 1 Georg Glaeser.
p. cm.
Includes bibliographical references and index.
1. Computer graphics. 2. Computer algorithms. 3. Three
dimensional display systems. 1. Tide.
T385.G575 1994
006.6'7-dc20 94-8076
ISBN 978-3-540-94288-7 ISBN 978-3-662-25798-2 (eBook)
DOI 10.1007/978-3-662-25798-2
Printed on acid-free paper.
©1994 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1994.
All rights reserved. This work may not be translated or copied in whole or in part without
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Production managed by Henry Krell; manufacturing supervised by Jacqui Ashri.
Photocomposed copy created from author's LaTeX files.
9 8 7 6 5 4 3 2 (Second corrected printing, 1995)
Preface
In this book, a variety of algoritbms are described that may be of interest to
everyone who writes software for 3D-graphics. It is a book that haB been written
for programmers at an intermediate level as well aB for experienced software
engineers who simply want to have some particular functions at their disposal,
without having to think too much about details like special cases or optimization
for speed.
The programming language we use is C, and that has many advantages, because
it makes the code both portable and efficient. Nevertheless, it should be possible
to adapt the ideas to other high-level programming languages.
The reader should have a reasonable knowledge of C, because sophisticated pro
grams with economical storage household and fast sections cannot be written
without the use of pointers. You will find that in the long run it is just aB easy
to work with pointer variables as with multiple arrays .
.Aß the title of the book implies, we will not deal with algorithms that are very
computation-intensive such as ray tracing or the radiosity method. Furthermore,
objects will always be (closed or not closed) polyhedra, which consist of a certain
number of polygons.
Ray tracing algorithms are necessary to get highly realistic pictures, and it is
even possible to make ray-traced movies out of complicated scenes. If you want
to do that, however, you will need the fastest computers available on the market
and an enormous amount of disk space. In order to create hundreds of frames on
less sophisticated computers, we need different algorithms. The loss of realism is
more than offset by the increase in speed.
vi Preface
The method of the book is to introduce theoretical background and program
ming code at the same time. Type definitions, global variables and macros are
described before their first applications. Because each chapter of the book builds
upon the previous one, it is not advisable to skip any of them. If you are already
familiar with a particular topic, at least skim over the relevant pages. All global
variables, macros and function prototypes are listed in the index of the book.
In summary, it may be said that the intention of this book is to
• provide a mathematical background for 3D-graphics.
• gradually develop a complete graphics program that is able to render images
of 3D-scenes comparatively quickly, including shadows and - to a limited
degree - reßections. The images can be stored as PostScript files. The scene
can be animated either interactively or by animation files. Series of scenes
can be stored and replayed in real time. The program can be ported to any
computer that is able to create palette colors by means of RGB-values and
set pixels in those colors on the screen. The C compiler should provide a
function to draw a line between two screen points and if possible a function
to fill convex polygons.
• illustrate techniques of C programming with emphasis on portability and
speed. Readers who are not so familiar with the programming language C
may also find this book useful for a better understanding of pointer arith
metic.
• provide the reader with a source code of a graphics programming package.
With the help of this source code, it is possible to adapt new code and
to implement new information. (For example, one can easily introduce new
spline types or new families of surfaces.)
Acknowledgments
A book like this can never be the achievement of one person only. It is not easy
to start with anyone, because many people and institutions have helped to get
the work done.
The idea of writing the book was born several years aga during the time I worked
with Steve M. Slaby at Princeton University. Besides Steve, the man who came
in with the most essential innovations and ideas from the very beginning was
Silvio Levy from the Mathematics Department. It was he who strongly believed
in the project.
At the same time, I received a lot of support from Dave Dobkin from the Com
puter Science Department and the individuals from the Interactive Computer
Graphics Laboratory.
Preface vii
Back in Europe, Reinhard Thaller, Leonid Dimitrov and Emanuel Wenger from
the Austrian Academy of Science helped me in many ways. During the same
time, I worked together with two of my students, Thomas Grohser and Hein
rich Pommer, both of whom brought in a lot of ideas to improve the code and
the speed. FUrthermore, I received advice from Hellmuth Stachel and Wolfgang
Rath from the Institute of Geometry at the Technical University of Vienna and
Otto Prem from the Handelsakademie St. Johann im Pongau, who also did the
proofreading together with Silvio Levy.
Suggested Reading
The list of books on 3D-graphics is already very long and it is useless to recom
mend them all. The books listed below may be useful for a better understanding
of this book:
• The C Programming Language: [DARN88j, [KERN86j .
• Theory of 3D-Graphics: [HEAR86j, [HOSC89j, [NEWM84j, [pLAS86j,
[ROGE85j, [ROGE90j, [THAL87j, [FOLE90j, [VINC92j.
Contents
1 A Basic Course in Spatial Analytic Geometry 1
1.1 Vectors . . . . · ... · ... · .... 1
1.2 How to Measure Lengths and Angles . . . . .... 9
1.3 Intersections of Lines and Planes · ... · . . . . 11
1.4 'franslations . . . · ......... · .... 18
1.5 Matrices .. .... · .... · ... · .... 20
1.6 Rotations ..... · ... · .... · . · .... 22
2 Projections 25
2.1 Central Projections . · ... 26
2.2 The Viewing Pyramid · .... 29
2.3 Coordinate Systems · ........ 32
2.4 Back and Forth Between Coordinate Systems · .... 37
2.5 Clipping Algorithms · ... . . . . . · ...... 43
3 How to Describe Three-Dimensional Objects 57
3.1 Data Pools . . . · .... · ... · . . ... 58
3.2 The 'Polyhedron' and 'Face' Structures .. 66
3.3 General Convex Objects · ... · ... · .... 68
X Contents
3.4 Surfaces of Revolution . . . 74
3.5 Surfaces of Translation . . . 82
3.6 The Intersection of Objects 85
4 Graphics Output 95
4.1 Graphics Hardware. 95
4.2 System-Dependent Macros and Functions .............. . 96
4.3 How to Create Color Palettes and How to Use Them . . . . . . . . . 99
4.4 Wire Frames and Depth Cuing . . . . . . . 105
4.5 Shading...................... 110
4.6 Basic Graphics Output Algorithms 117
5 A Fast Hidden-Surface Algorithm 125
5.1 Objects with Convex Outlines. . . . . . . . 127
· ........
5.2 Surfaces of Revolution 129
5.3 Sliced Surfaces . . . . · ........ 131
5.4 Function Graphs .. · ........ 133
5.5 Priority Among Objects · ........ 135
..........
5.6 Final Priority List 143
5.7 The Creation of Object Groups . . . . . . . . 149
5.8 Bounding Boxes and Separating Planes 150
6 Advanced Features 155
6.1 Convex Hulls ................ . 156
6.2 The Intersection of Convex Hulls ..... . 164
6.3 Shadows....... . ....... . 172
6.4 Reßections................... 177
6.5 Patterns.................... 180
6.6 Refraction, Transparent Objects 182
7 Hidden-Line Removal 185
7.1 A Quick Screen-Oriented Method ................... . 185
7.2 Hidden-Line Removal on Objects with Convex Outlines ...... . 186
7.3 Depth Buffering for Visibility . . . . . . . . . . . . . . . . . . . . . . 194
Contents xi
8 Mathematical Curves and Surfaces 201
8.1 Parametrized Curves ........ . 202
8.2 Classes of Parametrized Surfaces . . 205
8.3 Surfaces Given by Implicit Equations . 210
8.4 Special Curves on Mathematical Surfaces 212
8.5 A More Sophisticated Illumination Model 228
8.6 Shadow Buffering for Shadows .. 231
9 Spline Curves and Spline Surfaces 233
9.1 Interpolating Curves and Surfaces 233
9.2 Approximating Curves and Surfaces 242
9.3 Special Curves on Polygonized Surfaces 247
9.4 Spatial Integral Curves ...... . 249
9.5 SomeExamples of Applications . 253
10 Computer-Generated Movies 255
10.1 An Economical Way of Storing a Movie 255
10.2 A Fast Movie Previewer ......... . 266
11 The Programming Package on Your Disk 267
11.1 The Contents of the Disk ..... . 267
11.2 How to Instali the Program Package 268
11.3 How to Use the Program ...... . 269
11.4 How to Write Data Files and Animation Files . 271
11.5 How Fast Is Your Computer? ......... . 271
12 System Dependencies and Other Programming Languages 279
12.1 How to Change the System-Dependent Commands 279
12.2 'Macromania': C, ANSI C or C++ ? 283
12.3 Pointer Arithmetic in PASCAL 284
References 289
Index 293
