December 13, 2010 Time: 06:48pm fm.tex Discrete and Computational GEOMETRY This page intentionally left blank December 13, 2010 Time: 06:48pm fm.tex Discrete and Computational GEOMETRY SATYAN L. DEVADOSS and JOSEPH O’ROURKE PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD iii December 13, 2010 Time: 06:48pm fm.tex Copyright c 2011 by Princeton University Press Requests for permission to reproduce material from this work should be sent to Permissions, Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Devadoss, Satyan L., 1973– Discrete and computational geometry / Satyan L. Devadoss and Joseph O’Rourke. p. cm. Includes index. ISBN 978-0-691-14553-2 (hardcover : alk. paper) 1. Geometry–Data processing. I. O’Rourke, Joseph. II. Title. QA448.D38D48 2011 516.00285–dc22 2010044434 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Typeset by S R Nova Pvt Ltd, Bangalore, India press.princeton.edu Printed in China 10987654321 iv December 13, 2010 Time: 06:48pm fm.tex SLD Dedication. To my family, for their unconditional love of this ragamuffin. JOR Dedication. In memory of Michael Albertson. v This page intentionally left blank December 13, 2010 Time: 06:48pm fm.tex Contents Preface ix 1 POLYGONS 1.1 Diagonals and Triangulations 1 1.2 Basic Combinatorics 7 1.3 The Art Gallery Theorem 13 1.4 Scissors Congruence in 2D 20 1.5 Scissors Congruence in 3D 26 2 CONVEX HULLS 2.1 Convexity 33 2.2 The Incremental Algorithm 36 2.3 Analysis of Algorithms 39 2.4 Gift Wrapping and Graham Scan 42 2.5 Lower Bound 46 2.6 Divide-and-Conquer 48 2.7 Convex Hull in 3D 51 3 TRIANGULATIONS 3.1 Basic Constructions 59 3.2 The Flip Graph 66 3.3 The Associahedron 73 3.4 Delaunay Triangulations 79 3.5 Special Triangulations 87 4 VORONOI DIAGRAMS 4.1 Voronoi Geometry 98 4.2 Algorithms to Construct the Diagram 104 4.3 Duality and the Delaunay Triangulation 107 4.4 Convex Hull Revisited 113 5 CURVES 5.1 Medial Axis 118 5.2 Straight Skeleton 125 5.3 Minkowski Sums 128 5.4 Convolution of Curves 132 December 13, 2010 Time: 06:48pm fm.tex viii CONTENTS 5.5 Curve Shortening 138 5.6 The Heat Equation 144 5.7 Curve Reconstruction 148 6 POLYHEDRA 6.1 Platonic Solids 156 6.2 Euler’s Polyhedral Formula 162 6.3 The Gauss-Bonnet Theorem 170 6.4 Cauchy Rigidity 177 6.5 Shortest Paths 188 6.6 Geodesics 200 7 CONFIGURATION SPACES 7.1 Motion Planning 206 7.2 Polygonal Chains 215 7.3 Rulers and Locked Chains 221 7.4 Polygon Spaces 229 7.5 Particle Collisions 237 Appendix: Computational Complexity 245 Permissions 249 Index 251 December 13, 2010 Time: 06:48pm fm.tex Preface Although geometry is as old as mathematics itself, discrete geometry only fully emerged in the twentieth century, and computational geometry was only christened in the late 1970s. The terms “discrete” and “computational” fit well together, as the geometry must be discretized in preparation for computations. “Discrete” here means concentration on finite sets of points, lines, triangles, and other geometric objects, and is used to contrast with “continuous” geometry, for example, smooth sur- faces. Although the two endeavors were growing naturally on their own, it has been the interaction between discrete and computational geometry that has generated the most excitement, with each advance in one field spurring an advance in the other. The interaction also draws upon two traditions: theoretical pursuits in pure mathematics and applications- driven directions often arising in computer science. The confluence has made the topic an ideal bridge between mathematics and computer science. It is precisely to bridge that gap that we have written this book. In line with this goal, our presentation is sprinkled with both algo- rithms and theorems, with sometimes the theorem serving as the main thrust (e.g., the Gauss-Bonnet theorem), and sometimes an algorithm the primary goal of a section and theorems playing a supporting role (e.g., the flip graph computation of the Delaunay triangulation). As our emphasis is on the geometry of the subject, the algorithms presented in this book are strongly rooted in geometric intuition and insight. We describe the algorithms independent of any particular programming language, and in fact we do not even employ pseudocode, trusting that our boxed descriptions can be read as code by those steeped in the computer science idiom. Thus, no programming experience is needed to read this book. Algorithm complexities are discussed using the big-Oh notation without an assumption of prior exposure to this style of thinking, which is (lightly) covered in the Appendix. We include many proofs that we feel would interest a mathematically inclined student, presenting them in what we hope is an accessible style. At many junctures we connect to more advanced concerns, not fearing, for example, to jump to higher dimensions to make a relevant remark. At the same time, we connect each topic to applications that were often the initial motivation for studying the topic. Although we include careful December 13, 2010 Time: 06:48pm fm.tex x PREFACE proofs of theorems, we also try to develop intuition through visualization. Geometry demands figures! Some exposure to proofs is needed to gain that mystical “mathematical maturity.” We invoke calculus only in a few sections. A course in discrete mathematics is the more relevant prerequisite, but any course that presents formal proofs of theorems would suffice, such as linear algebra or automata theory. The material should be completely accessible to any mathematics or computer science major in the second or third year of college. In order to reach interesting advanced topics without the careful preparation they often demand, we sometimes offer a proof sketch (always marked as such), instead of a long, detailed formal proof of a result. Here we try to convince the reader that a formal proof is likely to be possible by sketching in the outlines without the details. Whether the reader can imagine those details from the outline is a measure of mathematical experience. A parallel skill of “computa- tional maturity” is needed to imagine how to implement our algorithm descriptions. The book is studded with Exercises, which we have chosen to place wherever they are relevant, rather than gather them at the end of each chapter. Some merely test a grasp of the foregoing material, most require more substantive thought (suitable for homework assignments), and starred exercises are difficult, often connecting to a published paper. A solutions manual is available to instructors from the publisher. Rather than include a scholarly bibliography, we have opted instead for “Suggested Readings” at the end of each chapter, providing pointers for further investigation. Between these pointers and websites such as Wikipedia, the reader should have no difficulty exploring the vast area beyond our coverage. And what lies beyond is indeed vast. The Handbook of Discrete and Computational Geometry runs to 1,500 pages and even so is highly compressed. Our coverage represents a sparse sampling of the field. We have chosen to cover polygons, convex hulls, triangulations, and Voronoi diagrams, which we believe constitute the core of discrete and computational geometry. Beyond this core, there is considerable choice, and we have selected several topics on curves and polyhedra, concluding with configuration spaces. The selection is skewed to the research interests of the authors, with perhaps more coverage of associahedra (first author) and unfolding (second author) than might be chosen by a committee of our peers. At the least, this ensures that we touch on the frontiers of current research. Because of the relative youth of the field, there are many accessible unsolved problems, which we highlight throughout. Although some have resisted the assaults of many talented researchers and may be awaiting a theoretical breakthrough, others may be accessible with current techniques and only await significant attention by an enterprising reader. The field has expanded greatly since its origins, and the new con- nections to areas of mathematics (such as algebraic topology) and new December 13, 2010 Time: 06:48pm fm.tex PREFACE xi application areas (such as data mining) seems only to be accelerating. We hope this book can serve to open the door on this rich and fascinating subject. Acknowledgments. Vickie Kearn was the ideal editor for us: firm but kind, and unfailingly enthusiastic. A special thanks goes to wise Jeff Erickson, who read the entire manuscript in draft, corrected many errors, suggested many exercises, and in general educated us in our own specialties to a degree we did not think possible. We are humbled and grateful. Satyan Devadoss: To all my students at Williams College who have learned this beautiful subject alongside me, while enduring my brutal exams, I am truly grateful. I especially thank Katie Baldiga, Jeff Danciger, Thomas Kindred, Rohan Mehra, Nick Perry, and Don Sheehy for being on the front line with me, with special thanks to Tomio Ueda for literally laying the foundation to this book. I am indebted to my colleagues and mentors Colin Adams, Mike Davis, Tamal Dey, Peter March, Jack Morava, Frank Morgan, Alan Saalfeld, and Jim Stasheff, all of whom have given generously of their time and wisdom over the years.