Table Of Content

Geometry and Computing Series Editors HerbertEdelsbrunner LeifKobbelt KonradPolthier EditorialAdvisoryBoard Jean-DanielBoissonnat GunnarCarlsson BernardChazelle Xiao-ShanGao CraigGotsman LeoGuibas Myung-SooKim TakaoNishizeki HelmutPottmann RobertoScopigno Hans-PeterSeidel SteveSmale PeterSchröder DietrichStoyan Forfurthervolumes: www.springer.com/series/7580 • Efi Fogel • Dan Halperin • Ron Wein CGAL Arrangements and Their Applications A Step-by-Step Guide 123 EfiFogel RonWein TelAvivUniversity TelAvivUniversity TheBlavatnikSchoolofComputerScience TheBlavatnikSchoolofComputerScience SchreiberBuilding SchreiberBuilding 69978TelAviv 69978TelAviv Israel Israel efi[email protected] [email protected] DanHalperin TelAvivUniversity TheBlavatnikSchoolofComputerScience SchreiberBuilding 69978TelAviv Israel [email protected] ISSN1866-6795 e-ISSN1866-6809 ISBN978-3-642-17282-3 e-ISBN978-3-642-17283-0 DOI10.1007/978-3-642-17283-0 SpringerHeidelbergDordrechtLondonNewYork LibraryofCongressControlNumber: 2011945423 ACMComputingClassification:I.3.5,F.2.2 MathematicsSubjectsClassification(2000):65Dxx,68U05 (cid:2)c Springer-VerlagBerlinHeidelberg2012 Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallythe rightsoftranslation, reprinting, reuseofillustrations, recitation, broadcasting,reproductiononmicrofilmorinanyotherway,and storageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable toprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc. inthispublicationdoesnotimply,evenintheabsenceofa specificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Ilil, Adam, Shira, and Tom. E.F. To Gili and Naomi. D.H. To Hélène, Nogah, and Shir. R.W. • Contents Preface xi What This Book Contains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi What You Should Know before Reading This Book . . . . . . . . . . . . . . . . . . . . . xi How to Obtain and Install Cgal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Style Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 1 Introduction 1 1.1 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Generic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Concepts and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Traits Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.3 Generic and Object-Oriented Programming . . . . . . . . . . . . . . . . . . 5 1.2.4 Libraries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Geometric Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Separation of Topology and Geometry . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Exact Geometric Computation . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Well-Behaved Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 The Computational Geometry Algorithm Library . . . . . . . . . . . . . . . . . . . 10 1.4.1 Cgal Chronicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.2 Cgal Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.3 The Algebraic Foundations of Cgal . . . . . . . . . . . . . . . . . . . . . . 11 1.4.4 The Geometry Kernel of Cgal . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.5 The State of Cgal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.6 The Namespace of Cgal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Basic Arrangements 19 2.1 Representation of Arrangements: The Dcel . . . . . . . . . . . . . . . . . . . . . . 19 2.2 The Main Arrangement Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Traversing the Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.2 Modifying the Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Input/Output Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Application: Obtaining Silhouettes of Polytopes . . . . . . . . . . . . . . . . . . . 36 2.4 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 vii viii Contents 3 Queries and Free Functions 43 3.1 Issuing Queries on an Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Point-Location Queries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.2 Vertical Ray Shooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Two Algorithmic Frameworks: Plane Sweep and Zone Construction . . . . . . . . . 48 3.3 Batched Point-Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Free Insertion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.1 Incremental Insertion Functions . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 Aggregate Insertion Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5 Removing Vertices and Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 Vertical Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6.1 Application: Decomposing an Arrangement of Line Segments . . . . . . . 59 3.7 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 Arrangements of Unbounded Curves 67 4.1 Representing Arrangements of Unbounded Curves . . . . . . . . . . . . . . . . . . 67 4.1.1 Basic Manipulation and TraversalMethods . . . . . . . . . . . . . . . . . . 69 4.1.2 Free Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Point-Line Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.1 Application: Minimum-Area Triangle . . . . . . . . . . . . . . . . . . . . 74 4.2.2 A Note on the Input Precision . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.3 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Arrangement-Traits Classes 83 5.1 The Hierarchy of Traits-Class Concepts . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1.1 The Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1.2 Supporting Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1.3 Supporting Arbitrary Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1.4 The Landmark Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.1.5 Supporting Unbounded Curves . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.6 The Traits Adaptor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Traits Classes for Line Segments and Linear Objects . . . . . . . . . . . . . . . . . 91 5.2.1 The Caching Segment-Traits Class . . . . . . . . . . . . . . . . . . . . . . . 92 5.2.2 The Non-Caching Segment-Traits Class . . . . . . . . . . . . . . . . . . . . 92 5.2.3 The Linear-Traits Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 The Polyline-Traits Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.4 Traits Classes for Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4.1 A Traits Class for Circular Arcs and Line Segments . . . . . . . . . . . . . 98 5.4.2 A Traits Class for Conic Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4.3 A Traits Class for Arcs of Rational Functions . . . . . . . . . . . . . . . . . 105 5.4.4 A Traits Class for Planar Bézier Curves . . . . . . . . . . . . . . . . . . . . 109 5.4.5 A Traits Class for Planar Algebraic Curves of Arbitrary Degree . . . . . . . 111 5.5 Traits-Class Decorators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.6 Application: Polygon Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.7 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6 Extending the Arrangement 129 6.1 The Notification Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Extending the Dcel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.2.1 Extending the Dcel Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.2.2 Extending All the Dcel Records . . . . . . . . . . . . . . . . . . . . . . . . 134 Contents ix 6.2.3 Input/Output for Arrangements with Auxiliary Data. . . . . . . . . . . . . 137 6.3 OverlayingArrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 Storing the Curve History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4.1 Traversing an Arrangement with History. . . . . . . . . . . . . . . . . . . . 147 6.4.2 Modifying an Arrangement with History . . . . . . . . . . . . . . . . . . . . 148 6.4.3 Input/Output for Arrangements with Curve History . . . . . . . . . . . . . 151 6.5 Application: Polygon Repairing and Winding Numbers . . . . . . . . . . . . . . . 152 6.6 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7 Adapting to Boost Graphs 161 7.1 The Primal Arrangement Representation. . . . . . . . . . . . . . . . . . . . . . . . 161 7.2 The Dual Arrangement Representation . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.3 Application: Largest Common Point Sets under (cid:2)-Congruence . . . . . . . . . . . 166 7.4 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8 Operations on (Curved) Polygons 175 8.1 Operations on Linear Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.1.1 Polygons with Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8.1.2 Operations on Polygons with Holes . . . . . . . . . . . . . . . . . . . . . . . 181 8.1.3 Point Containment and Non-Regularized Operations . . . . . . . . . . . . . 183 8.1.4 Connecting Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.1.5 Operations on Polygon Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.1.6 A Sequence of Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.1.7 Inserting Non-Intersecting Polygons . . . . . . . . . . . . . . . . . . . . . . 187 8.1.8 Performing Multiway Operations . . . . . . . . . . . . . . . . . . . . . . . . 189 8.2 Operations on Curved Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.2.1 The Traits-Class Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 8.2.2 Operations on Polygons with Circular Arcs . . . . . . . . . . . . . . . . . . 192 8.2.3 General-PolygonSet Traits-Adaptor . . . . . . . . . . . . . . . . . . . . . . 194 8.3 Application: Multiway Operations on General Polygons . . . . . . . . . . . . . . 198 8.4 Application: Obtaining Silhouettes of Polyhedra . . . . . . . . . . . . . . . . . . 203 8.5 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 9 Minkowksi Sums and Offset Polygons 209 9.1 Computing the Minkowski Sum of Two Polygons . . . . . . . . . . . . . . . . . . . 209 9.1.1 Computing Minkowski Sum Using Convolutions. . . . . . . . . . . . . . . . 211 9.1.2 Decomposition Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 9.2 Offsetting a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 9.2.1 Approximating the Offset with a Guaranteed Error Bound. . . . . . . . . . 215 9.2.2 Computing the Exact Offset. . . . . . . . . . . . . . . . . . . . . . . . . . . 217 9.3 Motion-Planning Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 9.3.1 Application: A Translating PolygonalRobot . . . . . . . . . . . . . . . . 220 9.3.2 Application: Coordinating Two Disc Robots . . . . . . . . . . . . . . . . . 227 9.4 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 10 Envelopes 241 10.1 Envelopes of Curves in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.1.1 Representing the Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.1.2 Constructing the Envelope Diagram . . . . . . . . . . . . . . . . . . . . . . 242 10.1.3 The Envelope-Software Components . . . . . . . . . . . . . . . . . . . . . . 243 x Contents 10.1.4 Using the Traits Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 10.2 Application: Nearest Jeep over Time . . . . . . . . . . . . . . . . . . . . . . . . . 248 10.3 Envelopes of Surfaces in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 10.3.1 The Envelope-Traits Concept . . . . . . . . . . . . . . . . . . . . . . . . . . 252 10.3.2 Using the Envelope-Traits Classes . . . . . . . . . . . . . . . . . . . . . . . 254 10.4 Application: Locating the Farthest Point. . . . . . . . . . . . . . . . . . . . . . . 257 10.5 Bibliographic Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11 Prospects 263 11.1 Arrangements on Curved Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 11.2 Higher-Dimensional Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 11.3 Fixed-Precision Geometric Computing . . . . . . . . . . . . . . . . . . . . . . . . . 267 11.4 More Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Bibliography 271 Index 285

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Arrangements of curves constitute fundamental structures that have been intensively studied in computational geometry. Arrangements have numerous applications in a wide range of areas – examples include geographic information systems, robot motion planning, statistics, computer-assisted surgery an
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Arrangements of curves constitute fundamental structures that have been intensively studied in computational geometry. Arrangements have numerous applications in a wide range of areas – examples include geographic information systems, robot motion planning, statistics, computer-assisted surgery an

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