Geometric Tools for Computer Graphics.pdf

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1 Introduction

How to Use This Book

Issues of Numerical Computation

Low-Level Issues

High-Level Issues

A Summary of the Chapters

2 Matrices and Linear Systems

Introduction

Motivation

Organization

Notational Conventions

Tuples

Definition 15

Arithmetic Operations

Matrices

Notation and Terminology

Transposition

Arithmetic Operations

Matrix Multiplication

Linear Systems

Linear Equations

Linear Systems in Two Unknowns

General Linear Systems

Row Reductions, Echelon Form, and Rank

Square Matrices 32

Diagonal Matrices

Triangular Matrices

The Determinant

Inverse

Linear Spaces

Fields

Definition and Properties

Subspaces

Linear Combinations and Span

Linear Independence, Dimension, and Basis

Linear Mappings

Mappings in General

Linear Mappings

Matrix Representation of Linear Mappings

Cramer’s Rule 50

Eigenvalues and Eigenvectors

Euclidean Space

Inner Product Spaces

Orthogonality and Orthonormal Sets

Least Squares 56

Further Reading

Appendix C Basic Formulas for Geometric Primitives

Introduction

Triangles

C.2.1 Symbols

C.2.2 Definitions

C.2.3 Right Triangles

C.2.4 Equilateral Triangle

C.2.5 General Triangle

Quadrilaterals

C.3.1 Square

C.3.2 Rectangle

C.3.3 Parallelogram

C.3.4 Rhombus

C.3.5 Trapezoid

C.3.6 General Quadrilateral

Circles

C.4.1 Symbols

C.4.2 Full Circle

C.4.3 Sector of a Circle

C.4.4 Segment of a Circle

Polyhedra

C.5.1 Symbols

C.5.2 Box

C.5.3 Prism

C.5.4 Pyramid

Cylinder

Cone

Spheres

C.8.1 Segments

C.8.2 Sector

Torus 960

Geometric Tools for Computer Graphics

The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling Series Editor: Brian A. Barsky, University of California, Berkeley Geometric Tools for Computer Graphics Philip Schneider and David Eberly Level of Detail for 3D Graphics David Luebke, Martin Reddy, Jonathan D. Cohen, Amitabh Varshney, Benjamin Watson, and Robert Huebner Texturing & Modeling: A Procedural Approach, Third Edition David S. Ebert, F. Kenton Musgrave, Darwyn Peachey, Ken Perlin, and Steven Worley Jim Blinn’s Corner: Notation, Notation, Notation Jim Blinn Understanding Virtual Reality William Sherman and Alan Craig Digital Video and HDTV Algorithms and Interfaces Charles Poynton Pyramid Algorithms: A Dynamic Programming Ap- proach to Curves and Surfaces for Geometric Modeling Ron Goldman Non-Photorealistic Computer Graphics: Modeling, Rendering, and Animation Thomas Strothotte and Stefan Schlechtweg Curves and Surfaces for CAGD: A Practical Guide, Fifth Edition Gerald Farin Subdivision Methods for Geometric Design: A Constructive Approach Joe Warren and Henrik Weimer Computer Animation: Algorithms and Techniques Rick Parent The Computer Animator’s Technical Handbook Lynn Pocock and Judson Rosebush Andrew Glassner’s Notebook: Recreational Computer Graphics Andrew S. Glassner Warping and Morphing of Graphical Objects Jonas Gomes, Lucia Darsa, Bruno Costa, and Luiz Velho Jim Blinn’s Corner: Dirty Pixels Jim Blinn Rendering with Radiance: The Art and Science of Lighting Visualization Greg Ward Larson and Rob Shakespeare Introduction to Implicit Surfaces Edited by Jules Bloomenthal Jim Blinn’s Corner: A Trip Down the Graphics Pipeline Jim Blinn Interactive Curves and Surfaces: A Multimedia Tutorial on CAGD Alyn Rockwood and Peter Chambers Wavelets for Computer Graphics: Theory and Applica- tions Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin Principles of Digital Image Synthesis Andrew S. Glassner Radiosity & Global Illumination Fran¸cois X. Sillion and Claude Puech Knotty: A B-Spline Visualization Program Jonathan Yen User Interface Management Systems: Models and Algorithms Dan R. Olsen, Jr. Making Them Move: Mechanics, Control, and Animation of Articulated Figures Edited by Norman I. Badler, Brian A. Barsky, and David Zeltzer Advanced RenderMan: Creating CGI for Motion Pictures Anthony A. Apodaca and Larry Gritz Geometric and Solid Modeling: An Introduction Christoph M. Hoffmann Curves and Surfaces in Geometric Modeling: Theory and Algorithms Jean Gallier An Introduction to Splines for Use in Computer Graphics and Geometric Modeling Richard H. Bartels, John C. Beatty, and Brian A. Barsky

Geometric Tools for Computer Graphics Philip J. Schneider David H. Eberly

Publishing Director Diane Cerra Publishing Services Manager Edward Wade Senior Developmental Editor Belinda Breyer Project Management Elisabeth Beller Cover Design Ross Carron Design Cover Image Getty/Spencer Jones Text Design Rebecca Evans & Associates Composition Windfall Software, using ZzTEX Technical Illustration and Figure Revision Dartmouth Publishing, Inc. Copyeditor Ken DellaPenta Proofreader Indexer Printer The Maple-Vail Book Manufacturing Group Jennifer McClain Steve Rath Designations used by companies to distinguish their products are often claimed as trademarks or registered trademarks. In all instances in which Morgan Kaufmann Publishers is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. Morgan Kaufmann Publishers An imprint of Elsevier Science 340 Pine Street, Sixth Floor San Francisco, CA 94104-3205 www.mkp.com © 2003 by Elsevier Science (USA) All rights reserved Printed in the United States of America 07 06 05 04 03 5 4 3 2 1 No part of this publication 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 the publisher. Library of Congress Control Number: 2002107242 ISBN: 1-55860-594-0 This book is printed on acid-free paper.

To my wife, Suzanne, and my sons, Dakota and Jordan —PS To my wife, Shelly, for her patience through yet another book —DE

Foreword Eric Haines On my shelf is an old book called A Programmer’s Geometry, by Bowyer and Wood- wark. It was published in 1983, reprinted twice in 1984 and 1985, but then discontin- ued. Over the years I have guarded my copy, keeping careful track of who borrowed it. Checking on the Web today, I found six used copies ranging in price from $50 to $100. This is a high price range for a paperback book only 140 pages in length. The reason the book is worth this much is that it describes how to program various op- erations related to 2D geometry. It does not just present geometric formulae; it also describes efﬁcient ways to accomplish tasks and gives code snippets (in FORTRAN). Now, almost two decades later, we have a worthy successor to that slim volume. The book before you distills a huge amount of literature on geometry into that which is most useful to programmers. The ﬁeld of computer graphics has evolved consid- erably since 1983, and this volume reﬂects those advances. Due to the continuing improvement in computer processor performance, operations that once were only part of ofﬂine analysis packages are now commonly done in interactive programs. Polygon triangulation, collision detection and response, and surface modelling and modiﬁcation are now possible at real-time rates. This book gives solid explanations and code to perform these and many other algorithms. Beyond providing a solid reference for a wide range of geometry-related tasks, this volume also presents the underpinnings of the theory behind the algorithms. Rather than employ a pure cookbook approach, which can leave the reader with runnable code and no comprehension of how it works, the authors explain key concepts. This approach makes each algorithm a tool that, further on, can be recombined with other tools. The dynamic nature of computer graphics makes it a particularly interesting area of study. Research and implementation of rendering methods respond to changes in the underlying hardware. For example, in the ﬁeld of interactive rendering, the emerging programmable nature of the graphics accelerator has changed the relative costs of different techniques. On a broader scale, the evolution of the CPU has made memory access and caching rise in importance, compared to the older practice of minimizing the number of operations (e.g., counting multiplies and adds). However, the underlying theory and algorithms for, say, ﬁnding the convex hull of an object are considerably more long-lasting, less affected by changes. Of course, more efﬁcient algorithms are found over time, and hardware inﬂuences which method currently is considered the fastest, but the basic principles remain the same. Years after you have vii

viii Foreword shed your books on DirectX 9 or Intel’s 64-bit Itanium architecture, you are likely to have some edition of this book on your shelf. Another reason this book will have increased staying power is the Internet. I am the archivist for the “Graphics Gems” series code repository. The code for this series of books, including code by Philip Schneider, was wisely made free for reuse when the series was published in the early 1990s. Over the years readers have sent in bug ﬁxes and improvements to the code base, so beneﬁting all. Similarly, Dave Eberly has carefully maintained his “Magic Software” Web site (www.magic-software.com), which includes working versions of many of the algorithms presented in this volume. Called “a national treasure” by a leading researcher in computer graphics, this site allows addenda and corrigenda to be made available instantly whenever they are needed. Code does not rust; it improves with age when properly supported. This is particularly true for algorithms in this book as they are not tied to particular hardware, network protocols, or other transient objects. Over the years I and many others have used algorithms and code by the authors in products and research projects. An hour of a programmer’s time often costs more than the price of a book. By this measure, you hold a volume potentially worth thousands of dollars. That it can be purchased for a fraction of this cost I consider a modern miracle. The amount of information crammed into this book is incredible. The mathematics may be slow going at times, but the alternative would be to include wordier and less precise descriptions of fewer algorithms. If you are looking for a lightweight text you can read through and check off your list, keep searching. This book sometimes requires effort and struggle to fully comprehend but then, so do most of the worthwhile things in the world.

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