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S E C O N D E D I T I O N Handbook of Discrete and Computational Geometry © 2004 by Chapman & Hall/CRC DISCRETE_MATH-ROSEN Series .fh8 3/8/04 11:47 AM Page 1 DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. AT&T Laboratories Middletown, New Jersey Miklos Bona, Combinatorics of Permatations Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan Gross and Jay Yellen, Graph Theory and Its Applications Jonathan Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications with Maple Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Fundamental Number Theory with Applications © 2004 by Chapman & Hall/CRC DISCRETE_MATH-ROSEN Series .fh8 3/8/04 11:47 AM Page 2 Richard A. Mollin, An Introduction to Cryptography Richard A. Mollin, Quadratics Richard A. Mollin, RSA and Public-Key Cryptography Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Douglas R. Stinson, Cryptography: Theory and Practice, Second Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography © 2004 by Chapman & Hall/CRC ADVISORY EDITORIAL BOARD Bernard Chazelle Princeton University David P. Dobkin Princeton University Herbert Edelsbrunner Duke University Ronald L. Graham University of California, San Diego Victor Klee University of Washington Donald E. Knuth Stanford University Ja(cid:19)nos Pach City College, City University of New York Richard Pollack Courant Institute, New York University Gu(cid:127)nter M. Ziegler Technische Universit(cid:127)at Berlin © 2004 by Chapman & Hall/CRC S E C O N D E D I T I O N Handbook of Discr ete and Computational Geometry edited by Jacob E. Goodman Joseph O’Rourke CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2004 by Chapman & Hall/CRC C3014 disclaimer.fm Page 1 Thursday, March 11, 2004 1:35 PM Library of Congress Cataloging-in-Publication Data Handbook of discrete and computational geometry / edited by Jacob E. Goodman and Joseph O’Rourke. p. cm. — (The CRC Press series on discrete mathematics and its applications) Includes bibliographical references and index. ISBN 1-58488-301-4 (alk. paper) 1. Combinatorial geometry—Handbooks, manuals, etc. 2. Geometry—Data processing— Handbooks, manuals, etc., I. Goodman, Jacob E. II. O’Rourke, Joseph. III. Title IV. Series. QA167.H36 2004 516'.13—dc22 2004040662 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-301-4/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2004 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-301-4 Library of Congress Card Number 2004040662 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper © 2004 by Chapman & Hall/CRC PREFACE While books and journals of high quality have proliferated in discrete and computational geometry during recent years, there has been to date no single reference work fully accessible to the nonspecialist as well as to the specialist, covering all the major aspects of both (cid:12)elds. The Handbook of Discrete and Computational Geometry is intended to do exactly that: to make the most important results and methods in these areas of geometry readily accessible to those who use them in their everyday work, both in the academic world|as researchers in mathematics and computer science|and in the professional world|as practitioners in (cid:12)elds as diverse as operations research, molecular biology, and robotics. A signi(cid:12)cant part of the growth that discrete mathematics as a whole has experienced in recent years has consisted of a substantial development in discrete geometry. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young (cid:12)eld of computational geometry. This synthesis between discrete and computational geometry, in which the methods and insights of each (cid:12)eld have stimulated new understanding of the other, lies at the heart of this Handbook. The phrase \discrete geometry," which at one time stood mainly for the areas of packing, covering, and tiling, has gradually grown to include in addition such areas as combinatorial geometry, convex polytopes, and arrangements of points, lines,planes,circles,and othergeometricobjectsintheplaneandin higherdimen- sions. Similarly, \computational geometry," which referred not long ago to simply the design and analysis of geometric algorithms, has in recent years broadened its scope,andnowmeansthestudyofgeometricproblemsfromacomputationalpoint ofview,includingalsocomputationalconvexity,computationaltopology,andques- tions involving the combinatorial complexity of arrangements and polyhedra. It is clear from this that there is now a signi(cid:12)cant overlap between these two (cid:12)elds, and in fact this overlaphas become one of practice aswell, asmathematicians and computerscientistshavefoundthemselvesworkingonthesamegeometricproblems and have forged successful collaborations as a result. At the same time, a growing list of areas in which the results of this work are applicable has been developing. It includes areas as widely divergent as engineering, crystallography, computer-aided design, manufacturing, operations research, geographicinformationsystems, robotics,error-correctingcodes, tomography,geometric modeling, computer graphics, combinatorialoptimization, computer vision, pattern recognition, and solid modeling. Withthisinmind,ithasbecomeclearthatahandbookencompassingthemost important results of discrete and computational geometry would bene(cid:12)t not only the workers in these two (cid:12)elds, or in related areas such as combinatorics, graph theory, geometric probability, and real algebraic geometry, but also the users of this body of results, both industrial and academic. This Handbook is designed to (cid:12)ll that role. We believe it will prove an indispensable working tool both for researchers in geometry and geometric computing and for professionals who use geometric tools in their work. The Handbook covers a broad range of topics in both discrete and computational geometry, as well as in a number of applied areas. These include geometric datastructures,polytopesandpolyhedra,convexhullandtriangulationalgorithms, packing and covering, Voronoi diagrams, combinatorial geometric questions, com- © 2004 by Chapman & Hall/CRC viii J.E. Goodman andJ. O’Rourke putational convexity, shortest paths and networks, computational real algebraic geometry, geometric arrangements and their complexity, geometric reconstruction problems, randomization and de-randomization techniques, ray shooting, parallel computation in geometry, oriented matroids, computational topology, mathematical programming, motion planning, sphere packing, computer graphics, robotics, crystallography, and many others. A (cid:12)nal chapter is devoted to a list of available software. Results are presented in the form of theorems, algorithms, and tables, witheverytechnicaltermcarefullyde(cid:12)nedinaglossarythatprecedesthesectionin whichthetermis (cid:12)rstused. Therearenumerousexamplesand(cid:12)gurestoillustrate the ideas discussed, as well as a large number of unsolved problems. The main body of the volume is divided into six parts. The (cid:12)rst two, on combinatorial and discrete geometry and on polytopes and polyhedra, deal with fundamental geometric objects such as planar arrangements, lattices, and convex polytopes. The next section, on algorithms and geometric complexity, discusses these basic geometric objects from a computational point of view. The fourth and (cid:12)fth sections, on data structures and computational techniques, discuss various computational methods that cut across the spectrum of geometric objects, such as randomization and de-randomization, and parallel algorithms in geometry, as well as eÆcient data structures for searching and for point location. The sixth section, which is the longest in the volume, contains chapters on fourteen applica- tionsareasofbothdiscreteandcomputationalgeometry,includinglow-dimensional linear programming, combinatorial optimization, motion planning, robotics, computer graphics, pattern recognition, graph drawing, splines, manufacturing, solid modeling, rigidity of frameworks, scene analysis, error-correcting codes, and crystallography. Itconcludeswith a(cid:12)fteenth chapter,anup-to-the-minutecompilation of available software relating to the various areas covered in the volume. A com- prehensive index follows, which includes proper names as well as all of the terms de(cid:12)ned in the main body of the Handbook. A word about references. Because it would have been prohibitive to provide complete references to all of the many thousands of results included in the Hand- book,wehavetoalargeextentrestrictedourselvestoreferencesforeitherthemost important results, or for those too recent to have been included in earlier survey booksorarticles;fortherestwehaveprovidedannotatedreferencestoeasilyacces- siblesurveysoftheindividual subjects coveredinthe Handbook,whichthemselves contain extensive bibliographies. In this way, the reader who wishes to pursue an older result to its source will be able to do so. On behalf ofthe sixty-onecontributorsand ourselves,wewould liketo express ourappreciationtoall thosewhosecomments wereof greatvalue tothe authorsof thevariouschapters: PankajK.Agarwal,NogaAlon,BorisAronov,SaugataBasu, Margaret Bayer, Louis Billera, Martin Blu(cid:127)mlinger, Ju(cid:127)rgen Bokowski, B.F. Cavi- ness,BernardChazelle,DannyChen,XiangpingChen,Yi-JenChiang,EdmundM. Clarke, Kenneth Clarkson, Robert Connelly, Henry Crapo, Isabel Cruz, Mark de Berg,Jesu(cid:19)sDeLoera,GiuseppeDiBattista,MichaelDrmota,PeterEades,Ju(cid:127)rgen Eckho(cid:11), Noam D. Elkies, Eva Maria Feichtner, Ioannis Fudos, Branko Gru(cid:127)nbaum, Dan Halperin, EszterHargittai, Ulli Hund, Ju(cid:127)rgHu(cid:127)sler, Peter Johansson,Norman Johnson,AmyJosefczyk,GilKalai,GyulaK(cid:19)arolyi,KevinKlenk,W lodzimierzKu- perberg, Endre Makai, Jr., Ji(cid:20)r(cid:19)(cid:16) Matou(cid:20)sek, Peter McMullen, Hans Melissen, Bengt Nilsson, Michel Pocchiola, Richard Pollack, J(cid:127)org Rambau, Ju(cid:127)rgen Richter-Gebert, Allen D. Rogers, Marie-Fran(cid:24)coise Roy, Egon Schulte, Dana Scott, Ju(cid:127)rgen Sellen, MichaSharir,PeterShor,MaximMichailovichSkriganov,NeilJ.A.Sloane,Richard © 2004 by Chapman & Hall/CRC Preface ix P.Stanley,G(cid:19)ezaT(cid:19)oth, IoannisTollis, LaureenTreacy,Alexander Vardy,GertVeg- ter, Pamela Vermeer, Sini(cid:20)sa Vre(cid:19)cica, Kevin Weiler, Asia Ivi(cid:19)c Weiss, Neil White, Chee-Keng Yap, and Gu(cid:127)nter M. Ziegler. In addition, we would like to convey our thanks to the editors of CRC Press forhavingthevisiontocommissionthisHandbookaspartoftheirDiscrete Mathe- matics and Its Applications series;totheCRCsta(cid:11),fortheirhelpwith thevarious stages of the project; and in particular to Nora Konopka, with whom we found it a pleasure to work from the inception of the volume. Finally, we want to expressour sincere gratitude to our families: Josy, Rachel, and Naomi Goodman, and Marylynn Salmon and Nell and Russell O’Rourke, for their patience and forbearance while we were in the throes of this project. Jacob E. Goodman Joseph O’Rourke PREFACE TO THE SECOND EDITION ThissecondeditionoftheHandbook of Discrete and Computational Geometry rep- resentsasubstantialrevisionofthe(cid:12)rstedition, publishedsevenyearsearlier. The new edition has added over 500 pages, a growth by more than 50%. Each chapter hasbeenthoroughlyrevisedandupdated,andwehaveaddedthirteennewchapters. Theadditionalroompermitted theexpansionofthe curtailed bibliographiesof the (cid:12)rst edition, which often required citing other surveys to locate original sources. The new bibliographies make the chapters, insofar as is possible, self-contained. Most chapters have been revised by their original authors, but in a few cases new authors have joined the e(cid:11)ort. All together, taking into account the chapters new to this edition, the number of authors has grown from sixty-three to eighty-two. In the (cid:12)rst edition there was one index; now there are two: in addition to the Index of De(cid:12)ned Terms there is also an Index of Cited Authors, which includes everyone referred to by name in either the text or the bibliography of each chapter. The (cid:12)rst edition chapter on computational geometry software has been split into two chapters: one on the libraries LEDA and CGAL, the other on additional software. There are (cid:12)ve new chapters in the applications section: on algorithms for modeling motion, on surface simpli(cid:12)cation and 3D-geometry compression, on statisticalapplications,onGeographicInformationSystemsandcomputationalcar- tography,and onbiologicalapplicationsof computational topology. Therearenew chaptersoncollisiondetectionandonnearestneighborsinhigh-dimensionalspaces. Wehaveaddedmaterialonmeshgeneration,aswellasanewchapteroncurveand surface reconstruction,and new chapterson embeddingsof (cid:12)nite metric spaces,on polygonal linkages, and on geometric graph theory. Allofthesenewchapters,togetherwiththemanynewresultscontainedwithin the Handbook as a whole, attest to the rapid growth in the (cid:12)eld since preparation forthe(cid:12)rsteditionbeganadecadeago. Andasbefore,wehaveengagedtheworld’s leading experts in each area as our authors. Inadditiontothemanypeoplewhohelpedwiththepreparationofthevarious chapterscomprisingthe(cid:12)rst edition, manyofwhomonceagaingaveinvaluableas- sistancewiththepresentedition,wewouldalsoliketothankthefollowingonbehalf © 2004 by Chapman & Hall/CRC x J.E. Goodman andJ. O’Rourke ofboththeauthorsandourselves: NinaAmenta,DavidAvis,MichaelBaake,David Bremner, Herv(cid:19)e Br(cid:127)onnimann, Christian Buchta, Sergio Cabello, Yi-Jen Chiang, Mirela Damian, Douglas Dunham, Stefan Felsner, Lukas Finschi, Bernd G(cid:127)artner, Ewgenij Gawrilow, Daniel Hug, Ekkehard K(cid:127)ohler, Je(cid:11)rey C. Lagarias, Vladimir I. Levenshtein, Casey Mann, Matthias Mu(cid:127)ller-Hannemann, Rom Pinchasi, Marc E. Pfetsch, Charles Radin, Jorge L. Ram(cid:19)(cid:16)rez Alfons(cid:19)(cid:16)n, Matthias Reitzner, Thilo Schr(cid:127)oder, Jack Snoeyink, Hellmuth Stachel, Pavel Valtr, and Nikolaus Witte. We would alsolike to express our appreciationto Bob Stern, CRC’s Executive Editor,whogaveusessentiallyafreehandinchoosinghowbestto(cid:12)lltheadditional 500 pages that were allotted to us for this new edition, as well as to Christine Andreasen for her sharp eye and unfailing good humor. Jacob E. Goodman Joseph O’Rourke © 2004 by Chapman & Hall/CRC