Computational geometry: algorithms and applications

388 Pages·2008·2.21 MB·English

by  Prof. Dr. Mark de Berg

Computational Geometry
Third Edition
Mark de Berg · Otfried Cheong
Marc van Kreveld · Mark Overmars
Computational Geometry
Algorithms and Applications
Third Edition
123
Prof.Dr.MarkdeBerg Dr.MarcvanKreveld
DepartmentofMathematics DepartmentofInformation
andComputerScience andComputingSciences
TUEindhoven UtrechtUniversity
P.O.Box513 P.O.Box80.089
5600MBEindhoven 3508TBUtrecht
TheNetherlands TheNetherlands
[e-mail] [e-mail]
Dr.OtfriedCheong,ne´Schwarzkopf Prof.Dr.MarkOvermars
DepartmentofComputerScience DepartmentofInformation
KAIST andComputingSciences
Gwahangno335,Yuseong-gu UtrechtUniversity
Daejeon305-701 P.O.Box80.089
Korea 3508TBUtrecht
[e-mail] TheNetherlands
[e-mail]
ISBN 978-3-540-77973-5 e-ISBN 978-3-540-77974-2
DOI 10.1007/978-3-540-77974-2
ACMComputingClassification(1998):F.2.2,I.3.5
LibraryofCongressControlNumber:2008921564
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Preface
Computational geometry emerged from the field of algorithms design and
analysisinthelate1970s. Ithasgrownintoarecognizeddisciplinewithits
ownjournals,conferences,andalargecommunityofactiveresearchers. The
successofthefieldasaresearchdisciplinecanontheonehandbeexplained
fromthebeautyoftheproblemsstudiedandthesolutionsobtained,and,onthe
otherhand,bythemanyapplicationdomains—computergraphics,geographic
informationsystems(GIS),robotics,andothers—inwhichgeometricalgorithms
playafundamentalrole.
Formanygeometricproblemstheearlyalgorithmicsolutionswereeither
slowordifficulttounderstandandimplement. Inrecentyearsanumberofnew
algorithmictechniqueshavebeendevelopedthatimprovedandsimplifiedmany
ofthepreviousapproaches. Inthistextbookwehavetriedtomakethesemodern
algorithmicsolutionsaccessibletoalargeaudience. Thebookhasbeenwritten
asatextbookforacourseincomputationalgeometry,butitcanalsobeusedfor
self-study.
Structureofthebook. Eachofthesixteenchapters(excepttheintroductory
chapter)startswithaproblemarisinginoneoftheapplicationdomains. This
problem is then transformed into a purely geometric one, which is solved
usingtechniquesfromcomputationalgeometry. Thegeometricproblemandthe
conceptsandtechniquesneededtosolveitaretherealtopicofeachchapter.The
choiceoftheapplicationswasguidedbythetopicsincomputationalgeometry
we wanted to cover; they are not meant to provide a good coverage of the
applicationdomains. Thepurposeoftheapplicationsistomotivatethereader;
thegoalofthechaptersisnottoprovideready-to-usesolutionsforthem.Having
saidthis,webelievethatknowledgeofcomputationalgeometryisimportant
tosolvegeometricproblemsinapplicationareasefficiently. Wehopethatour
bookwillnotonlyraisetheinterestofpeoplefromthealgorithmscommunity,
butalsofrompeopleintheapplicationareas.
Formostgeometricproblemstreatedwegivejustonesolution,evenwhen
anumberofdifferentsolutionsexist. Ingeneralwehavechosenthesolution
thatiseasiesttounderstandandimplement. Thisisnotnecessarilythemost
efficientsolution. Wealsotookcarethatthebookcontainsagoodmixtureof
techniqueslikedivide-and-conquer,planesweep,andrandomizedalgorithms.
Wedecidednottotreatallsortsofvariationstotheproblems;wefeltitismore
importanttointroduceallmaintopicsincomputationalgeometrythantogive
moredetailedinformationaboutasmallernumberoftopics. v
PREFACE Severalchapterscontainoneormoresectionsmarkedwithastar. Theycon-
tainimprovementsofthesolution,extensions,orexplaintherelationbetween
variousproblems. Theyarenotessentialforunderstandingtheremainderofthe
book.
EverychapterconcludeswithasectionthatisentitledNotesandComments.
Thesesectionsindicatewheretheresultsdescribedinthechapteroriginated,
mentionothersolutions,generalizations,andimprovements,andproviderefer-
ences. Theycanbeskipped,butdocontainusefulmaterialforthosewhowant
toknowmoreaboutthetopicofthechapter.
Attheendofeachchapteranumberofexercisesisprovided. Theserange
fromeasyteststocheckwhetherthereaderunderstandsthematerialtomore
elaboratequestionsthatextendthematerialcovered. Difficultexercisesand
exercisesaboutstarredsectionsareindicatedwithastar.
Acourseoutline. Eventhoughthechaptersinthisbookarelargelyindepen-
dent,theyshouldpreferablynotbetreatedinanarbitraryorder. Forinstance,
Chapter2introducesplanesweepalgorithms,anditisbesttoreadthischapter
beforeanyoftheotherchaptersthatusethistechnique. Similarly,Chapter4
shouldbereadbeforeanyotherchapterthatusesrandomizedalgorithms.
Forafirstcourseoncomputationalgeometry,weadvisetreatingChapters1–
10inthegivenorder. Theycovertheconceptsandtechniquesthat,according
tous,shouldbepresentinanycourseoncomputationalgeometry. Whenmore
materialcanbecovered,aselectioncanbemadefromtheremainingchapters.
Prerequisites. The book can be used as a textbook for a high-level under-
graduatecourseoralow-levelgraduatecourse, dependingontherestofthe
curriculum. Readersareassumedtohaveabasicknowledgeofthedesignand
analysisofalgorithmsanddatastructures: theyshouldbefamiliarwithbig-Oh
notations and simple algorithmic techniques like sorting, binary search, and
balancedsearchtrees.Noknowledgeoftheapplicationdomainsisrequired,and
hardlyanyknowledgeofgeometry. Theanalysisoftherandomizedalgorithms
usessomeveryelementaryprobabilitytheory.
Implementations. The algorithms in this book are presented in a pseudo-
codethat,althoughratherhigh-level,isdetailedenoughtomakeitrelatively
easytoimplementthem. Inparticularwehavetriedtoindicatehowtohandle
degenerate cases, which are often a source of frustration when it comes to
implementing.
Webelievethatitisveryusefultoimplementoneormoreofthealgorithms;
it will give a feeling for the complexity of the algorithms in practice. Each
chaptercanbeseenasaprogrammingproject. Dependingontheamountof
timeavailableonecaneitherjustimplementtheplaingeometricalgorithms,or
implementtheapplicationaswell.
Toimplementageometricalgorithmanumberofbasicdatatypes—points,
lines,polygons,andsoon—andbasicroutinesthatoperateonthemareneeded.
vi Implementingthesebasicroutinesinarobustmannerisnoteasy,andtakesalot
oftime.Althoughitisgoodtodothisatleastonce,itisusefultohaveasoftware PREFACE
libraryavailablethatcontainsthebasicdatatypesandroutines. Pointerstosuch
librariescanbefoundonourWebsite.
Website. ThisbookisaccompaniedbyaWebsite,whichcontainsalistof
erratacollectedforeacheditionofthebook,allfiguresandthepseudocodefor
allalgorithms,aswellassomeotherresources. Theaddressis
http://www.cs.uu.nl/geobook/
YoucanalsousetheaddressgivenonourWebsitetosenduserrorsyou
havefound,oranyothercommentsyouhaveaboutthebook.
Aboutthethirdedition. Thisthirdeditioncontainstwomajoradditions: In
Chapter7,onVoronoidiagrams,wenowalsodiscussVoronoidiagramsofline
segmentsandfarthest-pointVoronoidiagrams. InChapter12,wehaveincluded
anextrasectiononbinaryspacepartitiontreesforlow-densityscenes,asan
introductiontorealisticinputmodels. Inaddition,alargenumberofsmalland
some larger errors have been corrected (see the list of errata for the second
editionontheWebsite). Wehavealsoupdatedthenotesandcommentsofevery
chaptertoincludereferencestorecentresultsandrecentliterature. Wehave
triednottochangethenumberingofsectionsandexercises,sothatitshouldbe
possibleforstudentsinacoursetostillusethesecondedition.
Acknowledgments. Writing a textbook is a long process, even with four
authors. Many people contributed to the original first edition by providing
usefuladviceonwhattoputinthebookandwhatnot,byreadingchaptersand
suggestingchanges,andbyfindingandcorrectingerrors. Manymoreprovided
feedbackandfounderrorsinthefirsttwoeditions. Wewouldliketothankallof
them,inparticularPankajAgarwal,HelmutAlt,MarshallBern,JitBose,Hazel
Everett,GeraldFarin,SteveFortune,Geert-JanGiezeman,MordecaiGolin,Dan
Halperin,RichardKarp,MatthewKatz,KlaraKedem,NelsonMax,JosephS.B.
Mitchell,Rene´ vanOostrum,Gu¨nterRote,HenryShapiro,SvenSkyum,Jack
Snoeyink,GertVegter,PeterWidmayer,CheeYap,andGu¨ntherZiegler. We
alsowouldliketothankSpringer-Verlagfortheiradviceandsupportduringthe
creationofthisbook,itsneweditions,andthetranslationsintootherlanguages
(atthetimeofwriting,Japanese,Chinese,andPolish).
FinallywewouldliketoacknowledgethesupportoftheNetherlands’Or-
ganizationforScientificResearch(N.W.O.)andtheKoreaResearchFounda-
tion(KRF).
January2008 MarkdeBerg
OtfriedCheong
MarcvanKreveld
MarkOvermars
vii
Contents
1 ComputationalGeometry 1
Introduction
1.1 AnExample: ConvexHulls 2
1.2 DegeneraciesandRobustness 8
1.3 ApplicationDomains 10
1.4 NotesandComments 13
1.5 Exercises 15
2 LineSegmentIntersection 19
ThematicMapOverlay
2.1 LineSegmentIntersection 20
2.2 TheDoubly-ConnectedEdgeList 29
2.3 ComputingtheOverlayofTwoSubdivisions 33
2.4 BooleanOperations 39
2.5 NotesandComments 40
2.6 Exercises 41
3 PolygonTriangulation 45
GuardinganArtGallery
3.1 GuardingandTriangulations 46
3.2 PartitioningaPolygonintoMonotonePieces 49
3.3 TriangulatingaMonotonePolygon 55
3.4 NotesandComments 59
3.5 Exercises 60
4 LinearProgramming 63
ManufacturingwithMolds
4.1 TheGeometryofCasting 64
4.2 Half-PlaneIntersection 66
4.3 IncrementalLinearProgramming 71
4.4 RandomizedLinearProgramming 76 ix
CONTENTS 4.5 UnboundedLinearPrograms 79
4.6* LinearProgramminginHigherDimensions 82
4.7* SmallestEnclosingDiscs 86
4.8 NotesandComments 89
4.9 Exercises 91
5 OrthogonalRangeSearching 95
QueryingaDatabase
5.1 1-DimensionalRangeSearching 96
5.2 Kd-Trees 99
5.3 RangeTrees 105
5.4 Higher-DimensionalRangeTrees 109
5.5 GeneralSetsofPoints 110
5.6* FractionalCascading 111
5.7 NotesandComments 115
5.8 Exercises 117
6 PointLocation 121
KnowingWhereYouAre
6.1 PointLocationandTrapezoidalMaps 122
6.2 ARandomizedIncrementalAlgorithm 128
6.3 DealingwithDegenerateCases 137
6.4* ATailEstimate 140
6.5 NotesandComments 143
6.6 Exercises 144
7 VoronoiDiagrams 147
ThePostOfficeProblem
7.1 DefinitionandBasicProperties 148
7.2 ComputingtheVoronoiDiagram 151
7.3 VoronoiDiagramsofLineSegments 160
7.4 Farthest-PointVoronoiDiagrams 163
7.5 NotesandComments 167
7.6 Exercises 170
8 ArrangementsandDuality 173
SupersamplinginRayTracing
8.1 ComputingtheDiscrepancy 175
8.2 Duality 177
8.3 ArrangementsofLines 179
x 8.4 LevelsandDiscrepancy 185
8.5 NotesandComments 186 CONTENTS
8.6 Exercises 188
9 DelaunayTriangulations 191
HeightInterpolation
9.1 TriangulationsofPlanarPointSets 193
9.2 TheDelaunayTriangulation 196
9.3 ComputingtheDelaunayTriangulation 199
9.4 TheAnalysis 205
9.5* AFrameworkforRandomizedAlgorithms 208
9.6 NotesandComments 214
9.7 Exercises 215
10 MoreGeometricDataStructures 219
Windowing
10.1 IntervalTrees 220
10.2 PrioritySearchTrees 226
10.3 SegmentTrees 231
10.4 NotesandComments 237
10.5 Exercises 239
11 ConvexHulls 243
MixingThings
11.1 TheComplexityofConvexHullsin3-Space 244
11.2 ComputingConvexHullsin3-Space 246
11.3*TheAnalysis 250
11.4*ConvexHullsandHalf-SpaceIntersection 253
11.5*VoronoiDiagramsRevisited 254
11.6 NotesandComments 256
11.7 Exercises 257
12 BinarySpacePartitions 259
ThePainter’sAlgorithm
12.1 TheDefinitionofBSPTrees 261
12.2 BSPTreesandthePainter’sAlgorithm 263
12.3 ConstructingaBSPTree 264
12.4*TheSizeofBSPTreesin3-Space 268
12.5 BSPTreesforLow-DensityScenes 271
12.6 NotesandComments 278
12.7 Exercises 279 xi