Handbook of Discrete and Computational Geometry

1842 Pages·2018·17.961 MB·english

by  Csaba D. Toth,‎ Joseph O’Rourke,‎ Jacob E. Goodman

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DISCRETE MATHEMATICS AND ITS APPLICATIONS
Handbook of
DISCRETE AND
COMPUTATIONAL
GEOMETRY
THIRD EDITION
Edited by
Jacob E. Goodman
Joseph O’Rourke
Csaba D. Tóth
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CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Goodman, Jacob E. | O’Rourke, Joseph. | Tóth, Csaba (Csaba D.)
Title: Handbook of discrete and computational geometry / [edited by] Jacob Goodman, Joseph O’Rourke, Csaba Tóth.
Description: Third edition. | Boca Raton : CRC Press, 2017. | Includes bibliographical references.
Identifiers: LCCN 2017017843 | ISBN 9781498711395
Subjects: LCSH: Combinatorial geometry--Handbooks, manuals, etc. |
Geometry--Data processing--Handbooks, manuals, etc.
Classification: LCC QA167 .H36 2017 | DDC 516/.13--dc23
LC record available at https://lccn.loc.gov/2017017843
Visit the Taylor & Francis Web site at
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and the CRC Press Web site at
http://www.crcpress.com
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TABLE OF CONTENTS
Prefaces ..............................................................................xi
Contributors .........................................................................xv
COMBINATORIAL AND DISCRETE GEOMETRY..................................1
1 Finite point con(cid:12)gurations (J. Pach)...............................................3
2 Packing and covering (G. Fejes T(cid:19)oth).............................................27
3 Tilings (E. Harriss, D. Schattschneider, and M. Senechal).........................67
4 Helly-type theorems and geometric transversals (A. Holmsen and R. Wenger).....91
5 Pseudoline arrangements (S. Felsner and J.E. Goodman).........................125
6 Oriented matroids (J. Richter-Gebert and G.M. Ziegler)..........................159
7 Lattice points and lattice polytopes (A. Barvinok) ...............................185
8 Low-distortion embeddings of (cid:12)nite metric spaces
(P. Indyk, J. Matou(cid:20)sek, and A. Sidiropoulos) ..............................211
9 Geometry and topology of polygonal linkages (R. Connelly and E.D. Demaine)...233
10 Geometric graph theory (J. Pach)................................................257
11 Euclidean Ramsey theory (R.L. Graham) ........................................281
12 Discrete aspects of stochastic geometry (R. Schneider)...........................299
13 Geometric discrepancy theory and uniform distribution
(J.R. Alexander, J. Beck, and W.W.L. Chen)..............................331
14 Polyominoes (G. Barequet, S.W. Golomb, and D.A. Klarner).....................359
POLYTOPES AND POLYHEDRA..................................................381
15 Basic properties of convex polytopes
(M. Henk, J. Richter-Gebert, and G.M. Ziegler)............................383
16 Subdivisions and triangulations of polytopes (C.W. Lee and F. Santos)...........415
17 Face numbers of polytopes and complexes (L.J. Billera and A. Bj(cid:127)orner)..........449
18 Symmetry of polytopes and polyhedra (E. Schulte)...............................477
19 Polytope skeletons and paths (G. Kalai) .........................................505
20 Polyhedral maps (U. Brehm and E. Schulte) .....................................533
COMBINATORIAL AND COMPUTATIONAL TOPOLOGY.......................549
21 Topological methods in discrete geometry (R.T. Z(cid:20)ivaljevi(cid:19)c).......................551
22 Random simplicial complexes (M. Kahle).........................................581
23 Computational topology of graphs on surfaces (E. Colin de Verdi(cid:18)ere).............605
24 Persistent homology (H. Edelsbrunner and D. Morozov)..........................637
25 High-dimensional topological data analysis (F. Chazal)...........................663
ALGORITHMS AND COMPLEXITY OF
FUNDAMENTAL GEOMETRIC OBJECTS......................................685
26 Convex hull computations (R. Seidel)............................................687
27 Voronoi diagrams and Delaunay triangulations (S. Fortune)......................705
28 Arrangements (D. Halperin and M. Sharir) ...................................... 723
29 Triangulations and mesh generation (M. Bern, J. Shewchuk, and N. Amenta) ....763
30 Polygons (J. O’Rourke, S. Suri, and C.D. T(cid:19)oth) .................................787
31 Shortest paths and networks (J.S.B. Mitchell)....................................811
32 Proximity algorithms (J.S.B. Mitchell and W. Mulzer)...........................849
33 Visibility (J. O’Rourke)..........................................................875
34 Geometric reconstruction problems (Y. Disser and S.S. Skiena)..................897
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x
35 Curve and surface reconstruction (T.K. Dey).....................................915
36 Computational convexity (P. Gritzmann and V. Klee)............................937
37 Computational and quantitative real algebraic geometry
(S. Basu and B. Mishra)....................................................969
GEOMETRIC DATA STRUCTURES AND SEARCHING..........................1003
38 Point location (J. Snoeyink).....................................................1005
39 Collision and proximity queries (Y. Kim, M.C. Lin, and D. Manocha)...........1029
40 Range searching (P.K. Agarwal).................................................1057
41 Ray shooting and lines in space (M. Pellegrini)..................................1093
42 Geometric intersection (D.M. Mount) ...........................................1113
43 Nearest neighbors in high-dimensional spaces (A. Andoni and P. Indyk).........1135
COMPUTATIONAL TECHNIQUES............................................... 1157
44 Randomizaton and derandomization
(O. Cheong, K. Mulmuley, and E. Ramos).................................1159
45 Robust geometric computation (V. Sharma and C.K. Yap)......................1189
46 Parallel algorithms in geometry (M.T. Goodrich and N. Sitchinava) .............1225
47 Epsilon-approximations and epsilon-nets (N. Mustafa and K. Varadarajan)......1241
48 Coresets and sketches (J. Phillips) .............................................. 1269
APPLICATIONS OF DISCRETE AND COMPUTATIONAL GEOMETRY........1289
49 Linear programming (M. Dyer, B. G(cid:127)artner, N. Megiddo, and E. Welzl)..........1291
50 Algorithmic motion planning (D. Halperin, O. Salzman, and M. Sharir).........1311
51 Robotics (D. Halperin, L.E. Kavraki, and K. Solovey)...........................1343
52 Computer graphics (D. Dobkin and S. Teller)...................................1377
53 Modeling motion (L.J. Guibas and M. Roeloffzen)...............................1401
54 Pattern recognition (J. O’Rourke and G.T. Toussaint) ..........................1421
55 Graph drawing (E. Di Giacomo, G. Liotta, and R. Tamassia)...................1451
56 Splines and geometric modeling (C.L. Bajaj)....................................1479
57 Solid modeling (C.M. Hoffmann and V. Shapiro)................................1503
58 Computation of robust statistics: Depth, median, and related measures
(P.J. Rousseeuw and M. Hubert)...........................................1541
59 Geographic information systems (M. van Kreveld)...............................1555
60 Geometric applications of the Grassmann-Cayley algebra (N.L. White)..........1581
61 Rigidity and scene analysis (B. Schulze and W. Whiteley).......................1593
62 Rigidity of symmetric frameworks (B. Schulze and W. Whiteley) ................1633
63 Global rigidity (T. Jorda(cid:19)n and W. Whiteley)....................................1661
64 Crystals, periodic and aperiodic (M. Senechal) ..................................1695
65 Applications to structural molecular biology (H. Edelsbrunner and P. Koehl)....1709
66 Geometry and topology of genomic (A.J. Blumberg and R. Rabad(cid:19)an)............1737
GEOMETRIC SOFTWARE........................................................1777
67 Software (M. Joswig and B. Lorenz).............................................1779
68 Two computational geometry libraries: LEDA and CGAL
(M. Hoffmann, L. Kettner, and S. N(cid:127)aher) .................................1799
Index of cited authors..............................................................1833
Index of de(cid:12)ned terms..............................................................1885
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PREFACE TO THE FIRST EDITION
While books and journals of high quality have proliferated in discrete and compu-
tational 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, andothergeometricobjectsintheplaneandinhigherdimen-
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 overlap has become one of practice as well, as mathematicians 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 engineer-
ing, crystallography, computer-aided design, manufacturing, operations research,
geographic information systems, robotics, error-correcting codes, tomography, geo-
metric modeling, computer graphics, combinatorial optimization, 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 computa-
tional 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-
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xii J.E. Goodman, J. O’Rourke, and C.D. T(cid:19)oth
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, mathemat-
ical 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
which the term is (cid:12)rst used. There are numerous examples and (cid:12)gures to illustrate
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 efficient 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, com-
puter graphics, pattern recognition, graph drawing, splines, manufacturing, solid
modeling, rigidity of frameworks, scene analysis, error-correcting codes, and crys-
tallography. Itconcludeswitha(cid:12)fteenthchapter,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-
siblesurveysoftheindividualsubjectscoveredintheHandbook, 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 of the sixty-one contributors and ourselves, we would like to express
our appreciation to all those whose comments were of great value to the authors of
thevariouschapters: PankajK.Agarwal,BorisAronov,NogaAlon,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)s de Loera, Giuseppe Di Battista, Michael Drmota, Peter Eades, Ju(cid:127)rgen
Eckhoff, Noam D. Elkies, Eva Maria Feichtner, Ioannis Fudos, Branko Gru(cid:127)nbaum,
Dan Halperin, Eszter Hargittai, Ulli Hund, Ju(cid:127)rg Hu(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
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Preface xiii
P. Stanley, G(cid:19)eza T(cid:19)oth, Ioannis Tollis, Laureen Treacy, Alexander Vardy, Gert Veg-
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; totheCRCstaff, fortheirhelpwiththevarious
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 express our 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
ThissecondeditionoftheHandbookofDiscreteandComputationalGeometryrep-
resentsasubstantialrevisionofthe(cid:12)rstedition, publishedsevenyearsearlier. The
new edition has added over 500 pages, a growth by more than 50%. Each chapter
hasbeenthoroughlyrevisedandupdated,andwehaveaddedthirteennewchapters.
The additional room permitted the expansion of the curtailed bibliographies of the
(cid:12)rst edition, which often required citing other surveys to locate original sources.
The new bibliographies make the chapters, in so far as is possible, self-contained.
Most chapters have been revised by their original authors, but in a few cases new
authors have joined the effort. All together, taking into account the chapters new
to this edition, the number of authors has grown from sixty-three to eighty-four.
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 chap-
ter. 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 on biological applications of computational topology. There are new
chaptersoncollisiondetectionandonnearestneighborsinhigh-dimensionalspaces.
Wehaveaddedmaterialonmeshgeneration,aswellasanewchapteroncurveand
surface reconstruction, and new chapters on embeddings of (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
chapters comprising the (cid:12)rst edition, many of whom once again gave invaluable
assistance with the present edition, we would also like to thank the following on
behalf of both the authors and ourselves: Pankaj Agarwal, David Avis, Michael
Baake, David Bremner, Herv(cid:19)e Br(cid:127)onnimann, Christian Buchta, Sergio Cabello,
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xiv J.E. Goodman, J. O’Rourke, and C.D. T(cid:19)oth
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, Jeffrey C. La-
garias, 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, Ju(cid:127)rgen Sellen, Thilo Schr(cid:127)oder, Jack Snoeyink, Hellmuth Stachel, Pavel
Valtr, Nikolaus Witte, and Chee Yap.
We would also like to express our appreciation to Bob Stern, CRC’s Executive
Editor,whogaveusessentiallyafreehandinchoosinghowbestto(cid:12)lltheadditional
500 pages that were allotted to us for this new edition.
Jacob E. Goodman
Joseph O’Rourke
PREFACE TO THE THIRD EDITION
This third edition of the Handbook of Discrete and Computational Geometry ap-
pears 20 years after the (cid:12)rst edition. A generation of researchers and practitioners
have learned their craft using the Handbook. Since the publication of the 2nd edi-
tion in 2004, the (cid:12)eld has grown substantially. New methods have been developed
tosolvelongstandingopenproblems,withmanysuccesses. Advancesintechnology
demanded new computational paradigms in processing geometric data.
Fifty-eight out of sixty-(cid:12)ve chapters from the second edition of the Handbook
have been revised and updated, many of them by the original authors, others in
collaboration with new co-authors. Ten new chapters have been added. Five of
the new chapters are devoted to computational topology and its applications. A
new chapter on proximity algorithms gives a comprehensive treatment of relative
neighbohood graphs and geometric spanners. New chapters on coresets, sketches,
"-nets, and "-approximations cover geometric methods to cope with large data.
Twonew chaptersexpandon therecentbreakthroughsin rigiditytheory. Wehope
the Handbook will remain a one-stop reference book, accessible to both specialists
and nonspecialists.
Wewouldliketothankthefollowing,onbehalfofthe103authorsandourselves,
for providing valuable feedback on the preliminary version of various chapters:
BernardoA(cid:19)brego,AbdoAlfakih,ImreB(cid:19)ar(cid:19)any,S(cid:127)orenLennartBerg,M(cid:19)onicaBlanco,
Prosenjit Bose, Mireille Bousquet-M(cid:19)elou, Kevin Buchin, Steve Butler, Sergio Ca-
bello,PabloCamara,GunnarCarlsson,CesarCeballos,JosephChan,DannyChen,
Siu-WingCheng,VincentCohen-Addad,M(cid:19)onikaCsik(cid:19)os,AntoineDeza,KevinEm-
mett, Jeff Erickson, Silvia Fern(cid:19)andez-Merchant, Moritz Firsching, Florian Frick,
Xavier Goaoc, Steven Gortler, Christian Haase, Bill Jackson, Bruno Jartoux, Iyad
Kanj, Matya Katz, Bal(cid:19)azs Keszegh, Hossein Khiabanian, Csaba Kir(cid:19)aly, Matias
Korman, Francis Lazarus, Michael Lesnick, Michael Mandell, Arnaud de Mesmay,
TobyMitchell,EranNevo,PatrickvanNieuwenhuizen,ArnauPadrol,ValentinPol-
ishchuk,HannesPollehn,DanielRosenbloom,RamanSanyal,PaulSeiferth,Michiel
Smid, Yannik Stein, Jean Taylor, Louis Theran, Dimitrios Thilikos, Haitao Wang,
Andrew Winslow, Richard Wolff, Ge Xia, and Sakellarios Zairis.
Jacob E. Goodman
Joseph O’Rourke
Csaba D. T(cid:19)oth
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Part I
COMBINATORIAL AND
DISCRETE GEOMETRY
1
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1 FINITE POINT CONFIGURATIONS
J(cid:19)anos Pach
INTRODUCTION
The study of combinatorial properties of (cid:12)nite point con(cid:12)gurations is a vast area
of research in geometry, whose origins go back at least to the ancient Greeks.
Since it includes virtually all problems starting with \consider a set of n points
in space," space limitations impose the necessity of making choices. As a result,
we will restrict our attention to Euclidean spaces and will discuss problems that
we (cid:12)nd particularly important. The chapter is partitioned into incidence problems
(Section 1.1), metric problems (Section 1.2), and coloring problems (Section 1.3).
1.1 INCIDENCE PROBLEMS
Inthissectionwewillbeconcernedmainlywiththestructureofincidencesbetween
a(cid:12)nitepointcon(cid:12)gurationP andasetof(cid:12)nitelymanylines(or,moregenerally,k-
dimensional (cid:13)ats, spheres, etc.). Sometimes this set consists of all lines connecting
theelementsofP. TheprototypeofsuchaquestionwasraisedbySylvester[Syl93]
more than one hundred years ago: Is it true that for any con(cid:12)guration of (cid:12)nitely
manypointsintheplane,notallonaline,thereisalinepassingthroughexactlytwo
points? This problem was rediscovered by Erd}os [Erd43], and affirmative answers
to this question was were given by Gallai and others [Ste44]. Generalizations for
circlesandconicsectionsinplaceoflineswereestablishedbyMotzkin[Mot51]and
Wilson-Wiseman [WW88], respectively.
GLOSSARY
Incidence: A point of con(cid:12)guration P lies on an element of a given collection of
lines (k-(cid:13)ats, spheres, etc.).
Simple crossing: Apointincidentwithexactlytwoelementsofagivencollection
of lines or circles.
Ordinary line: A line passing through exactly two elements of a given point
con(cid:12)guration.
Ordinary circle: Acirclepassingthroughexactlythreeelementsofagivenpoint
con(cid:12)guration.
Ordinary hyperplane: A (d(cid:0)1)-dimensional (cid:13)at passing through exactly d ele-
ments of a point con(cid:12)guration in Euclidean d-space.
Motzkin hyperplane: A hyperplane whose intersection with a given d-dimen-
sional point con(cid:12)guration lies|with the exception of exactly one point|in a
(d(cid:0)2)-dimensional (cid:13)at.
3
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