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Premium Quality Steiner Triangulations

Permanent Link: http://ufdc.ufl.edu/UFE0041048/00001

Material Information

Title: Premium Quality Steiner Triangulations
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Erten, Hale
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: acute, computational, delaunay, generation, geometry, mesh, refinement, steiner, triangulation, voronoi
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A complex geometric domain is required to be partitioned into a small set of simple and high-quality geometric elements for accurate numerical analysis. Given a two-dimensional input domain, we consider its quality triangulations as the discretization of the domain into non-overlapping triangles while each satisfies certain quality constraints. Related to this, many researchers studied two major problems. The first one is to generate triangulations with a lower bound on small angles, while the second is to generate triangulations with an upper bound on large angles. An upper bound on large angles does not imply a non-trivial lower bound on small angles. Here, we propose novel ideas to generate premium quality triangulations with significantly better simultaneous small and large angle bounds than the bounds provided by the existing methods in practice. Delaunay refinement is a well-known triangulation technique, which generates size-optimal good-quality Delaunay triangulations. Delaunay refinement algorithms aim to compute triangulations that have all angles at least alpha. However, in practice, these algorithms usually work for alpha less than or equal to 34 degrees and fail to terminate for larger constraint angles. We describe two novel ideas to improve the performance of Delaunay refinement algorithms. The first idea is an effective use of the Voronoi diagram and unifies previously suggested point insertion schemes together with a new strategy. The second idea which is, integration of a local relocation strategy into the refinement process, leads to a strategy that has significantly better performance than performing these two tasks independently. In addition to providing the same theoretical guarantees as the previous algorithms, our approach generates smaller triangulations and generally terminates for constraint angles up to 42 degrees. We also extend our ideas to accommodate simultaneous small and large angle constraints. Experimental study shows that our method works for small angles as high as 41 degrees, and large angles as low as 81 degrees. Hence, we provide a substantial improvement over the existing results, especially for applications where acute or non-obtuse triangulations without small angles are desired. Employing these ideas provide the first software for computing premium quality triangulations for complex geometric domains.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hale Erten.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ungor, Alper.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041048:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041048/00001

Material Information

Title: Premium Quality Steiner Triangulations
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Erten, Hale
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: acute, computational, delaunay, generation, geometry, mesh, refinement, steiner, triangulation, voronoi
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A complex geometric domain is required to be partitioned into a small set of simple and high-quality geometric elements for accurate numerical analysis. Given a two-dimensional input domain, we consider its quality triangulations as the discretization of the domain into non-overlapping triangles while each satisfies certain quality constraints. Related to this, many researchers studied two major problems. The first one is to generate triangulations with a lower bound on small angles, while the second is to generate triangulations with an upper bound on large angles. An upper bound on large angles does not imply a non-trivial lower bound on small angles. Here, we propose novel ideas to generate premium quality triangulations with significantly better simultaneous small and large angle bounds than the bounds provided by the existing methods in practice. Delaunay refinement is a well-known triangulation technique, which generates size-optimal good-quality Delaunay triangulations. Delaunay refinement algorithms aim to compute triangulations that have all angles at least alpha. However, in practice, these algorithms usually work for alpha less than or equal to 34 degrees and fail to terminate for larger constraint angles. We describe two novel ideas to improve the performance of Delaunay refinement algorithms. The first idea is an effective use of the Voronoi diagram and unifies previously suggested point insertion schemes together with a new strategy. The second idea which is, integration of a local relocation strategy into the refinement process, leads to a strategy that has significantly better performance than performing these two tasks independently. In addition to providing the same theoretical guarantees as the previous algorithms, our approach generates smaller triangulations and generally terminates for constraint angles up to 42 degrees. We also extend our ideas to accommodate simultaneous small and large angle constraints. Experimental study shows that our method works for small angles as high as 41 degrees, and large angles as low as 81 degrees. Hence, we provide a substantial improvement over the existing results, especially for applications where acute or non-obtuse triangulations without small angles are desired. Employing these ideas provide the first software for computing premium quality triangulations for complex geometric domains.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hale Erten.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ungor, Alper.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041048:00001


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Firstandforemost,Iwouldliketoexpressmysinceregratitudetomyadvisor,Dr.AlperUngor,forallofhisguidance,supportandpatiencethroughoutmystudiesandresearch.IamreallygratefulthatheintroducedmetoComputationalGeometryandinspiredmewithhisimmenseknowledgeandgreatenthusiasm.Itwasagreatprivilegetoworkunderhissupervision.Iwouldalsoliketothankmycommitteemembers,Dr.JayGopalakrishnan,Dr.JeffreyHo,Dr.SanjayRanka,andDr.MeeraSitharam,foralloftheirhelpandvaluablesuggestions.IamgratefultoallmyteachersinUniversityofFlorida,fromwhomIhavelearnedvariousimportantsubjects.Ishallacknowledgethatmostoftheworkpresentedinthisthesishasbeenappearedinvariouspublications.ThematerialsinChapter 3 havebeenpublishedintheproceedingsoftheSGP07:ACMSIGGRAPH/EurographicsSymposiumonGeometryProcessing[ 51 ]andlaterpublishedinSIAMJournalofScienticComputing[ 53 ].ThematerialsinChapter 4 ,havebeenpublishedintheproceedingsofthe19thCanadianComputationalGeometryConference[ 50 ]andthe6thInternationalSymposiumonVoronoiDiagrams[ 54 ]andsubmittedtothejournalAppliedNumericalMathematics.Iwouldliketothankallmynancialsupportersthroughoutmyentirestudy.UniversityofFloridaComputerandInformationScienceandEngineeringdepartmentandNationalScienceFoundationhavebeenfundingmyresearchandeducationforfouryears.IalsothankUniversityofFloridaOfceofAuditandComplianceReviewandAnesthesiologydepartmentfortheirpartialsupports.IwasdelightedtoworkwithDr.PatrickM.KnuppatSandiaNationalLaboratoriesandDr.KirkHalleratSolidworksCorporationduringmysummerinternshipsintheirgroupsandthankfulforleadingmetoworkondiverseresearchprojectsandgivingmenancialsupport.IalsothankDr.JorgPetersforprovidingmethelatteropportunity. 4

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page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 LISTOFSYMBOLS .................................... 12 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1Motivation .................................... 16 1.2PreviousWork ................................. 18 1.3ProblemStatement ............................... 20 1.4ContributionsofResearch ........................... 22 2BACKGROUND ................................... 25 2.1Triangulations .................................. 25 2.1.1DelaunayTriangulations ........................ 26 2.1.1.1ConstrainedDelaunaytriangulations ............ 28 2.1.1.2ConformingDelaunaytriangulations ............ 29 2.1.2VoronoiDiagrams ............................ 30 2.1.3QualityTriangulations ......................... 31 2.1.4SteinerTriangulations ......................... 32 2.2SteinerQualityTriangulationMethods .................... 33 2.2.1DelaunayRenement ......................... 33 2.2.2Quadtree-basedMethods ....................... 35 2.2.3AdvancingFrontMethods ....................... 36 2.2.4Circle-basedMethods ......................... 38 2.2.5Smoothing(MeshImprovement)Methods .............. 38 2.3SteinerQualityTriangulationswithAngleConstraints ............ 41 2.3.1NoSmallAngleTriangulations ..................... 43 2.3.1.1Delaunayrenement ..................... 43 2.3.1.2Otherrenementalgorithms ................ 45 2.3.2NoLargeAngleTriangulations ..................... 45 2.3.3Smoothing ................................ 47 3TRIANGULATIONSWITHLOCALLYOPTIMALSTEINERPOINTS ....... 50 3.1Introduction ................................... 50 3.1.1OurIdeaandContribution ....................... 52 6

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................................ 52 3.1.2.1Terminationproblem ..................... 53 3.1.2.2Meshsize ........................... 56 3.2LocallyOptimalSteinerPoints ......................... 56 3.3Analysis ..................................... 60 3.4RelocationandRenement .......................... 63 3.5Experiments .................................. 64 3.5.1Implementation ............................. 65 3.5.2DataSets ................................ 66 3.5.3PerformanceComparison ....................... 66 3.5.4ResultsonUniformTriangulations ................... 76 3.5.5Comparison:SmoothingMethods ................... 84 4TRIANGULATIONSWITHOUTSMALLANDLARGEANGLES ......... 87 4.1Introduction ................................... 87 4.1.1OurIdeaandContribution ....................... 87 4.1.2Motivation ................................ 89 4.2-petal,-petal,-wedge,-slab,[,]-slice ................ 90 4.3IntersectionofSlicesandFreeVertexRelocation .............. 93 4.4Experiments .................................. 96 4.4.1Implementation ............................. 96 4.4.2DataSetsandOutputVisualization .................. 96 4.4.3Performance .............................. 97 4.4.4Comparison:PreviousDelaunayRenementAlgorithms ...... 106 4.4.5Comparison:SmoothingMethods ................... 109 4.4.6ResultsonUniformTriangulations ................... 112 5DISCUSSIONS .................................... 114 5.1aCuteSoftware ................................. 114 5.1.1SoftwareWebsite ............................ 114 5.1.2DownloadandUsage ......................... 114 5.1.3Comparison:DifferencesandSimilaritieswithTriangleSoftware .. 115 5.2FutureWork ................................... 116 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 128 7

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Table page 3-1PerformanceofAlgorithm 3 withrespecttoothers ................ 67 3-2NumberofdifferenttypesofSteinerpointsusedbydifferentalgorithms .... 68 3-3PercentagevaluesofdifferenttypesofSteinerpointsusedbyAlgorithm 3 ... 71 4-1Aqualitativecomparisonofourmethodagainstthepreviousmethods ..... 87 4-2Performanceofthealgorithmforfourdifferentangleranges ........... 103 4-3ComparisonofourmethodwiththepreviousDelaunayrenementmethods .. 105 8

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Figure page 1-1DelaunaytriangulationandaSteinerqualitytriangulationofacomplexdomain. 21 2-1Typesofinputdomainswithpossibletriangulations ................ 26 2-2Delaunaytriangulationemptycircleproperty ................... 27 2-3Maximizingtheminimumangleandminimizingthemaximumangle ...... 28 2-4Delaunaytriangulationandthree-dimensionalconvexhullrelationship ..... 28 2-5ConstrainedandconformingDelaunaytriangulations ............... 29 2-6VoronoidiagramandthedualDelaunaytriangulationofapointset ....... 30 2-7Steinertriangulationsofdifferentinputdomains .................. 32 2-8Outputtriangulationswithdifferentqualities .................... 34 2-9Unbalancedandbalancedquadtrees ........................ 35 2-10Advancingfrontow ................................. 36 2-11Outputofdifferentrenementalgorithms ...................... 37 2-12Circle-basedtriangulationexample ......................... 38 2-13RelocatingapointinLaplaciansmoothing ..................... 39 2-14Applyingsmoothing ................................. 41 2-15TypesofSteinerpoints ................................ 44 2-16Outputofdifferenttypesofrenementalgorithms ................. 46 2-17Outputofdifferenttypesofsmoothingalgorithms ................. 49 3-1TheoriginalDelaunayrenementalgorithmvs.ouridea ............. 51 3-2Circumcenterinsertion ................................ 53 3-3Terminationproblem ................................. 55 3-4Terminationproblemonasimpledomain ..................... 56 3-5ClassicationoftypesofSteinerpoints ...................... 58 3-6LocallyoptimalSteinerpoint ............................ 59 3-7TypeII(A)and(B) 61 9

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..................... 63 3-9OutputoftheAlgorithm 3 forBoeingandoctagondatasets ........... 66 3-10OutputoftheAlgorithm 3 forItalyandbutterydatasets. ............ 69 3-11OutputoftheAlgorithm 3 forcookinglady,seahorseandChinadatasets ... 70 3-12OutputsizecomparisonplotsforCyprusandcrabdatasets ........... 73 3-13OutputsizecomparisonplotsforhexagonandNewYorkviewdatasets .... 74 3-14OutputsizecomparisonplotsforBoeingandFloridadatasets ......... 75 3-15OutputsizecomparisonexampleforBoeingdataset ............... 77 3-16Outputsizecomparisonexampleforalligatordataset .............. 77 3-17OutputsizecomparisonexampleforFloridadataset ............... 78 3-18Plotofaverageoutputsizeratio ........................... 78 3-19Plotofrunningtimes ................................. 79 3-20Outputtriangulationswithmaxareaconstraintforvariousdatasets ....... 80 3-21OutputtriangulationswithmaxareaconstraintforTexasdataset ........ 81 3-22ExtensiveplotoftheoutputperformanceratiobasedonmaxareaforPuertoRicodataset ..................................... 81 3-23Plotoftheoutputperformanceratiobasedonmaxarea ............. 82 3-24Outputsizecomparisonwithmaxareaconstraint ................. 83 3-25Comparingsmoothingalgorithmswithourmethod ................ 86 4-1AcutetriangulationofthedomainAcute ...................... 88 4-2Feasibleregionslice ................................. 91 4-3AdditionaltypesofSteinerpoints .......................... 92 4-4Relocatingafreevertex ............................... 93 4-5Non-convexintersectionregions .......................... 94 4-6Approximateconvexsupersetofasliceregion .................. 94 4-7Coloringscales .................................... 97 4-8Plotofoutputsizeforvaryingand 98 10

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99 4-10Plotofoutputsizeandtimeperformanceofthealgorithmbasedonaxedandavariable 100 4-11OutputtriangulationsofChinadatasetforvariousminimumandmaximumangleconstraints ................................... 101 4-12Outputtriangulationsofgoldshdatasetforvariousminimumandmaximumangleconstraints ................................... 102 4-13OutputoftheclassicalDelaunayrenementmethod ............... 107 4-14Outputofourmethodbasedonmaximumandminimumangleconstraints ... 108 4-15Sizecomparisonofalldatasets .......................... 109 4-16ComparingoutputsizesandanglehistogramsofpreviousclassicalDelaunayrenementalgorithmandours ........................... 110 4-17Outputtriangulationsofvariousalgorithms .................... 111 4-18PremiumqualityoutputtriangulationsforBoeingandhexagoninputdomainswiththeadditionalmaximumareaconstraint ................... 112 4-19PremiumqualityoutputtriangulationsforTurkeydatasetwithdifferentmaximumareaconstraints ................................... 113 5-1AlgorithmicowofTriangleandaCuteforbuffalodataset ............ 117 11

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PSLGPlanarstraightlinegraphCDTConstrainedDelaunaytriangulationCVTCentroidalVoronoitessellationODTOptimalDelaunaytriangulationmax-minMaximizingminimummin-maxMinimizingmaximumConstraintminimumangleTheoreticalboundonminimumanglePracticalboundonminimumangleConstraintmaximumangle@Boundary 12

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11 13 16 21 ].Engineeringapplicationscreatetriangulations(generatemeshes)astherststepofniteelementmethodfornumericalcomputations.Complexobjectsbecomecombinationofsimpleelementssatisfyingwell-denedcriteria.Discretizationisrequiredforsolvingcontinuousproblems,sothattargetfunctionscanbeapproximatedwithpiecewisepolynomialsonelements[ 13 ].Accuracyofsolutionsregardingdifferentialandintegralequationsaswellasspeeddependonthequalityofthetriangulation[ 18 111 112 ].Thus,underlyingelementsshouldsatisfysomesizeandshapeproperties,usuallypreferredtobeclosetoequilateraltriangles[ 95 ].Industrialimplementationsarealwayslookingformoreefcientalgorithms,whiletheyneedtriangulationswhicharesuitablefortheirspecicapplicationstoavoidcomputationalerrors.Sincedifferentapplicationsexpectdifferentqualitymetricsfromtriangulations,numberofrelatedproblemsincreasesrapidly.Forinstance,theycouldbelookingforvariousconstraintssuchasangle,weight,area,edgelength,heightandsoon[ 5 11 95 ].Inthiscontext,BabuskaandAziz[ 6 ]showedthatlargeanglesbut 15

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105 ]pointedoutthatlargeanglesandsmallanglesareconictingqualitymeasuresandshowedthatsmallanglesarebadforminimizingtheconditionnumberofstiffnessmatrix,althoughlargeanglesareacceptable.Sincetriangulationsarenecessaryforsolvingwide-rangeofengineeringandscienticproblems,evenasmallprogressonalgorithmswouldbesignicanttowardsgettingmoreaccurateresultsonnumericalcalculationswhilesavingenormousamountoftimeandmoney. 6 11 105 ].So,signicantimprovementsovertheexistingtriangulationalgorithmsandsoftwareaddressingthetraditionalqualitycriteria,suchassmallestangle,arealwayswelcome.Insomecases,thequalityconstraintbecomesanabsolutenecessityasittranslatestothetheoreticalguaranteesonthenumericalalgorithm[ 105 119 ].Asnewnumericalmethods(aredeveloped)addressingtheneedsofemergingapplication,thereisgreaterneedfortriangulationalgorithmsandsoftwareprovidingvariousnewqualitycriteriaguarantee.Triangulationsizeandspeed.Thesizeofatriangulation,thatisthenumberoftriangles,isakeyfactorintherunningtimeoftheapplicationalgorithmwhichusesthetriangulation.Amongtwotriangulationswiththesamequalitybound,theonewiththe 16

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68 69 ].Oneofthegoalsinthisstudyistoreducetheoutputsizeofthetriangulationalgorithm.Thisalsohelpsusimprovethespeedofthetriangulationprocess.Asaresult,wehaveadouble-gaininimprovingthenumericalsimulationtime.Suchanimprovementinsomeapplicationsmaybethekeyperformingreal-timesimulations.Software.Inadditiontodesignofalgorithmsandheuristics,havingtheirrobustimplementationsintheformofpopularsoftwareisessentialforindustrialandscienticpractice.Amongseveraldifferentmethods,amethodcanbedistinguishedbyitsstrongcontributionsanditssuitablesoftwareimplementation.Consequently,engineeringapplicationswillconsiderusingthismethodasapartoftheirnumericalcomputations.However,robustnessisveryimportant[ 40 ].Otherwise,eventhesignicantbenetsofthemethodcouldbeignored.Furthermore,improvinganexistingrobustimplementationofapopularalgorithmwillautomaticallyhaveaneffectonrelatedapplications.Alternativequalitycriteria.Decomposingcomplexstructuresintotriangles(triangulation)satisfyingcertainqualitycriteriaisanintegralpartofengineeringapplications.Asaresult,theyarealwayslookingforbetteralgorithmsinducingmoreaccuracybyimprovingqualitycriteriaoftriangulations.Formajorityofnumericalapplications,qualitycriterionisrelatedtoangleswithinatriangulation.Notonlythesmallestangle,butalsothelargestanglehassignicantimportanceforapplications[ 36 71 96 119 ].Largersmallestangleandsmallerlargestangletriangulationsarepreferredtoothers.Moreover,insomecasesacutenessoftriangulationisnecessaryforefciencyofalgorithms[ 119 ].Inadditiontoangle,totaledgelengthsofatriangulationorminimumtriangleareainatriangulationalsoplaysanimportantrole.Thus,typesofqualitycriteriaischangingaccordingtorelevantengineeringapplication.Dynamicmeshing.Numericalsimulationofcomplexdomainsoftenrequiregeometricmodeling(triangulation)ofanevolvinggeometry.Thetriangulationneedstoadoptboththechangingdomainboundary,andthechangingnumericalconditions.In 17

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11 ].Inaddition,otherimportantsurveyshavealsobeencompleted[ 5 16 43 44 104 116 ].Decomposingapolygonalregionintopieces(triangles)byitselfisuseful,withoutevenconsideringanyoptimization/qualityconstraint.Ithasseveralpracticalapplicationsforsolvingwell-knownreallifeproblems[ 33 ].However,majorityofcontemporarypracticalmethodsfocusonqualitytriangulations,sinceapproximationerrorsofanapplicationdependonqualitymeasurementsoftriangulations[ 6 ].Inparticular,DelaunaytriangulationsattractmanyresearchersduetotheirconcurrentoptimizationofseveralqualitymeasuresandthembeingplanardualsofVoronoidiagrams[ 26 28 29 31 68 69 84 99 103 ].Qualitytriangulationproblemsusuallyrequireaddingnewpoints,calledtheSteinerpoints,totheinputdomainsinordertoimprovequalityandsatisfythegivenqualityconstraints.Throughouttheyears,Steinerqualitytriangulationshavebeengeneratedbyseveraldifferentapproaches,suchasgrid-based,circle-based,quadtree-basedandotherapproaches[ 7 13 15 27 48 88 ].Ingeneral,wecallthemrenementtechniques, 18

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1-1 givestheDelaunaytriangulationandaSteinerqualitytriangulationofacomplexgeometricdomain.Amongtheseapproaches,inparticularforgeneratingSteinerqualitytriangulationswithaminimumanglecriterion,Delaunay-basedandquadtree-basedmethodsbecamewidespreadinuse.DelaunayrenementmethodsinsertSteinerpointsdependingontheunderlyinginitialDelaunaytriangulation[ 28 29 45 99 118 ].Ontheotherhand,quadtree-basedmethodsusewell-knowndatastructures,quadtrees,tondalocationforanewSteinerpoint[ 13 48 88 124 ].(ReferSection 2.2.1 andSection 2.2.2 formoreinformation.)Thesetechniqueshavebeenfoundationsforessentialsoftwareproducts.Forexample,Triangle[ 106 ]softwareimplementsseveralDelaunayrenementalgorithms[ 101 102 ],whereasanothertriangulationsoftwareQMG[ 89 ]implementsquadtreerenementformeshgeneration[ 88 ].QuadtreescauselargeroutputtriangulationsandarelimitedforsomequalitymeasuresincomparisontoDelaunay-basedmethods.Asaresult,Delaunayrenementapproachesareusuallypreferredoverquadtreerenementapproaches.Therefore,thisaffectsthefrequencyamountinusageofTrianglesoftwarewhichishigherthanQMG.ManyresearchershaveintroducedideastoimproveDelaunayrenementalgorithmsfromdifferentaspects,likesmallestanglebound,outputsize,timeperformance,andotherimportantefciencycriteria[ 28 45 51 68 99 118 ].Largestanglecriterionisknowntobeessentialformanyapplications,aswellassmallestangle.Havingnon-obtuseoracutetriangulationscouldbemandatoryforsomealgorithmsortheymightleadtomoreefcientversionsofthealgorithms,likesurfaceinterpolation,andmeshgenerationalgorithms[ 11 119 ].Althoughtherehasbeensomeworkonnon-obtusetriangulations[ 7 12 13 15 49 ]andacutetriangulations[ 13 22 67 70 79 120 ],ageneralalgorithmfornosmallnolargeangleoracutetriangulationshasnotbeenproposed.Moreover,theexistingalgorithmsremainlimitedintheir 19

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11 ].Afterhavinganinitialtriangulationofagiveninputdomain,itcanbeimprovedintermsofqualitybyippingedgesoradjustingmeshverticeswhichareallowedtomove.Inparticular,smoothingcouldbedonebysimplecomputations,asinLaplaciansmoothing,oritcouldrequireexpensiveoptimizationtechniquesappliedonthetriangulationgloballyorlocally[ 3 16 46 56 125 ].Inaddition,fewstudiescombinedrenementmethodswithlocalsmoothing,howeverthesearecomplexandinefcientcombinations[ 32 117 ].Heuristicmethodsincludingpolygonaldecomposition[ 12 ],advancingfronttechniques[ 11 17 73 77 78 81 ]andotherspossessvariousadvantagesdependingontheproblemtobesolved[ 11 75 ].However,theyremainlesspopularforqualitytriangulationscomparedtootherrenementmethods,unlessintegratedwithothereffectivetechniques[ 75 118 ].FurtherpreviousworkdetailsaregiveninChapter 2 20

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B C Figure1-1. A)U.S.A.inputdomain.B)Delaunaytriangulationofthedomain.C)Steinerqualitytriangulationofthedomainhavingallangles>20. 21

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3 andChapter 4 presentdetailedexplanationsofthemethodsandachievements. 28 45 99 104 118 ].ThisdenitionfollowsfromanoveluseoftheVoronoidiagram,whichisdualtoDelaunaytriangulation.ThenewruleautomaticallyselectsoneofthefourtypesofSteinerpoints,threeofwhich(circumcenters,sinks,off-centers)arepreviouslystudied.Wealsoextendthisdenitionforgenerating[,]-triangulations.ThisnewapplicationofVoronoistructurecouldinspiresolutionsforotherresearchproblems. 13 28 98 99 ],andmeshsmoothing,i.e.,adjustingmeshvertices[ 3 35 60 ],aretwoimportanttechniquesforimprovingthequalityoftriangulations.Thesetwomethodsareusuallyusedoneaftertheother(smoothingfollowedbyrenement)[ 46 ]butrarelyintegrated[ 32 76 117 ].Weproposeasimpleframeworkwhereasimplelocalsmoothing(pointrelocation)strategyisintegratedintotherenementalgorithm.Itiscertainthatmanyotherapplicationscouldbenetfromthisapproach.Notethat,ourmethodsaremorepowerfulthanapplyingpost-smoothingonagood-qualitytriangulationtoprovidepremiumqualitytriangulations. 22

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11 ]: 1-1 ). Denition1. 62 ],O(nloglogn)[ 115 ]andO(n)[ 23 ]timealgorithmsregardingsimplepolygonaldomains.Forslightlymodiedversionsofthesameproblem,differentresultshavebeenobtained.Forinstance,someofthevariationsareproventobeNP-complete[ 11 ]. 25

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Figure2-1. Typesofinputdomainswithpossibletriangulationsindashedlines.A)Pointset.B)Simplepolygon.C)Polygonwithahole.D)PSLG. Denition2. 2-2 )makesDelaunaytriangulationsthedualofVoronoiDiagrams.Moreimportantly,theminimumangleofthetriangulationismaximizedamongallothertriangulationsofagiventwo-dimensionalpointset[ 108 ].SeeFigure 2-3 .ManyO(nlogn)-timealgorithmshavebeenintroducedforndingDelaunaytriangulationsindifferentways.Thelistcontainsconvexhullbasedalgorithms[ 97 ],divideandconqueralgorithms[ 41 65 74 ],plane-sweepalgorithms(whichgenerateVoronoidiagrams)[ 57 58 ]andrandomizedincrementalalgorithms[ 64 ]. 26

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Figure2-2. Delaunaytriangulationemptycircleproperty:ThecircumcircleofanytriangleintheDelaunaytriangulationisempty(containsnoverticesofP).A)InputpointsetP.B)NotaDelaunaytriangulation(emptycirclepropertyisviolated).C)Delaunaytriangulation. Inpractice,thesemethodshavebeenimprovedbyseveralstudies.Moreover,parallelversionshavebeenproposedtoprovidemoreefcientimplementations[ 30 110 ].AdetailedsurveyandacomparativestudyonDelaunaytriangulationscanbefoundin[ 58 113 ]. 27

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Figure2-3. Maximizingtheminimumangleandminimizingthemaximumangle.denotesthemaximumangle,denotestheminimumangleinthetriangulation.A)Notalocallyoptimaledge(notaDelaunaytriangulation,butthemaximumangleisminimized).B)Alocallyoptimaledge(Delaunaytriangulation,theminimumangleismaximized,butnotviceversa.). ABC Figure2-4. Inconvexhullbasedalgorithms,amappingbetweentheinputverticesandthree-dimensionalspaceisestablishedasaparaboloid.Then,thelowerthree-dimensionalconvexhulloftheparaboloidisprojectedontotheplanetoobtaintheDelaunaytriangulationofthepointset. Denition4.

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Figure2-5. ConstrainedandconformingDelaunaytriangulationsofthegivenPSLG.A)InputPSLG.B)ConstrainedDelaunaytriangulation.C)ConformingDelaunaytriangulation. 11 ].ForPSLGtypeofinputdomains,Delaunaytriangulationsaredenedasconstrained,sincethegivenedgesarerequiredtobepreservedasasingleedgeinthetriangulation.Notethat,aconstrainedDelaunaytriangulationmightcontaintrianglesthatarenotDelaunay.However,theysatisfytheconstrainedDelaunaycondition(seeFigure 2-5 (B)).AfterasimpleO(n2)-timeedgeippingalgorithm,ChewproposedanO(nlogn)-timedivideandconqueralgorithmforndingCDTsingeneral[ 31 ].Forconvexpolygons,Aggarwaletal.gavealinear-timealgorithm[ 1 ]. Denition6. 2-5 (C)).Notethat,Steinerpointsaregenerallyusedforimprovingthemeshquality.Hence,algorithmsemployingDelaunaytriangulationcanbeseenas 29

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28 ]andRuppert's[ 99 ]Delaunayrenementalgorithms. Denition7. Voronoidiagram(solidlines)andthedualDelaunaytriangulation(dashedlines)ofpointsetgiveninFigure 2-2 Voronoidiagramsareusedforsolvingvarietyofproblems.Inparticular,nearestneighborquerieshavenumerousapplications,suchaslocation-databaseapplications,dataminingproblemsaswellaspolymerphysicstopics[ 4 11 69 ].ConvexhullproblemandDelaunaytriangulationproblemscanbereducedtoVoronoidiagrams.Thus,algorithmsgeneratingVoronoidiagramsarealsoapplicableforDelaunaytriangulationsandviceversa.Simply,VoronoidiagramscanbegeneratedinO(n2logn)-timebyusinghalfplaneintersectionalgorithm.However,O(nlogn)-timealgorithmsareknownforndingtheDelaunaytriangulationofapointset.Hence,thismustbetrueforVoronoidiagramsaswell.Indeed,Fortune'sO(nlogn)-timesweeplinealgorithmturnsouttobe 30

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57 58 ].AnextensivesurveyonVoronoidiagramscanbefoundin[ 4 ]. Denition8. 100 ],convexhull[ 97 ],sweepline[ 57 58 ],andrandomizedincremental[ 64 ]approaches.Inaddition,edgeipping,edgeinsertionalgorithms,greedyanddynamicprogrammingmethodsarealsoused[ 11 ].Certainvariationsremainstillasopenproblemshavingonlyapproximateorheuristicsolutions,suchasminimumweighttriangulation.Adetailedstudyonoptimaltriangulationalgorithmscanbefoundin[ 121 ].Amongmanyoptimaltriangulationproblem,twoofthemaremorerelevanttoourstudythanothers.Thesearereferredasmax-minandmin-maxangletriangulations.Asnoted,Delaunaytriangulationsmaximizetheminimumangleinatriangulationandprovideadirectsolutiontomax-minangletriangulationproblem.Althoughavoiding 31

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2-3 .)Forndingsuchtriangulations,edgeippingmethodhasbeenshowntoprovidegoodresultsinpractice[ 47 63 ].However,thistechniquemightresultinalocaloptimumratherthanaglobalone.Hence,thecomputationallyexpensiveedgeinsertionalgorithmisintroduced[ 10 ]andhasbeenshowntoexactlysolvethemin-maxangletriangulationproblem[ 47 ]. Denition9. 11 ](seeFigure 2-7 ).Theseadditionalpoints,asmentionedbefore,arecalledSteinerpoints.AlthoughSteinerpointsmightimprovethequality,toomanypointscancauseproblemsorinefciencyforseveralapplicationsduetosize.However,thisnewtypeofqualitytriangulationissuitableformanyengineeringproblemsallowingextravertices,suchasmeshgenerationofastructuralobject. Figure2-7. TypesofinputdomainswithpossibleSteinertriangulationsindashedlines.A)Pointset.B)Simplepolygon.C)Polygonwithahole.D)PSLG. Notethat,Steinerversionsofqualitytriangulationproblemsneedanadditionalgoalassizeoptimalitytoavoidoversizedtriangulations,thatis,thereisatrade-offbetweenqualityandnumberofSteinerpoints.Forexample,theDelaunaytriangulation 32

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33 63 ].WhenSteinerpointsareallowed,theproblembecomesaSteinerqualitytriangulationproblemtondthesmallest-sizedtriangulationwithsmallanglelessthanathresholdvalue.However,thistypeofoptimalitymakestheproblemscomplex.Therefore,approximationalgorithmsprovidingqualitytriangulationsusingmoderatenumberofSteinerpointsareintroduced.MostcrucialpartoftheSteinertriangulationalgorithmsistohaveaneffectivepointinsertionstrategy.TherearenumerousstrategiesforlocatingSteinerpoints,includingDelaunay-based,grid-based,circle-basedandquadtree-basedmethods[ 7 13 15 27 48 88 ].Section 2.2 describesthesemethodsindetail. 11 ].Later,FreyintroducedtheideaofusingDelaunaytriangulationoftheinputandthenewboundaryvertices,andinsertSteinerpointsinsidethedomaintoeliminatebadtriangles,i.e.,non-qualitytriangles[ 61 ].ThebasicstrategyistokeeptheDelaunaypropertybyaddingcircumcentersofthebadtriangles.Sincethisisaninexpensivelocaloperation,Delaunayrenementalgorithmsbecamepopular.Notethat,therelationship 33

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2-8 ). ABC Figure2-8. Outputtriangulationswithdifferentqualities(:minimumangle).A)Delaunaytriangulation:=1.35,0Steinerpoints.B)Steinertriangulation:=20,106Steinerpoints.C)Steinertriangulation:=30,299Steinerpoints. ChewproposedaDelaunayrenementalgorithmtoobtainuniformdensitytriangulations[ 28 ].Hesimplyproposedtoinsertnearbypointsinordertoeliminatepoorqualitytriangleswherecircumcentersofthesetrianglesareusedasthelocationforthenewpoint[ 66 ].Then,RuppertintroducedmaybetherstDelaunayrenementalgorithmwhichgivesbothsizeandshapeguarantees[ 99 ]byfollowingasimilaridea.Heprovedthat,thealgorithmsuccessfullygeneratessize-optimalmesheswithnoanglessmallerthanaround20.7.Chewpublishedanotheralgorithmndingtriangulationswithsize-optimalityandnoanglessmallerthanaround26.5[ 29 ].Withoutsize-optimalityguarantee,theangleboundaryisadvancedto30byaspherepackingargument.TheclassicalDelaunayrenementalgorithmscanbegeneralizedasinAlgorithm 1 .However,theDelaunayrenementmethodstillsuffersfromtheso-calledterminationproblem[ 53 ],suchthatthealgorithmdoesnotterminateforlargeangles.PavalsostudiedDelaunayRenementalgorithmsinhisthesisandpointedoutsomecaseswheretheDelaunayrenementalgorithmfails[ 93 ].Inpractice,thismethodgeneratesSteinerqualitytriangulationsforminimumangleboundupto34.Asanalternativetocircumcenters,mainlysinks[ 45 ]andoff-centers[ 118 ]wereintroduced.Additional 34

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2.3 ComputetheDelaunaytriangulationoftheinput Denition10. 11 ].Grid-basedmethodsforSteinerqualitytriangulationshasbeenusedtoguaranteesomequalitymeasures[ 11 ].Forinstance,in[ 7 ],Bakeretal.useduniformgridstogiveboundsfortheanglesinthetriangulation.Quadtreescanbeseenasgridswithlevels(seeFigure 2-9 (B)),whichhasbeenusedtosolvequalitytriangulationproblems. ABC Figure2-9. A)Inputpointset.B)Unbalancedquadtree.C)Balancedquadtree. ForSteinerqualitytriangulations,quadtree-basedrenementalgorithmsinsertSteinerpointsbasedonthequadtreeformedfromtheinputdomain.Theygivetriangulationswithinaconstantfactoroftheoptimalandsatisfyvariousqualitymeasures[ 13 27 48 88 124 ].Inaddition,mostofthealgorithmsusebalanced 35

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2-9 (C)). 102 ].Also,theygenerateaxis-alignedtriangulationsandhavelimitationsonanglesandoncomputingboundaryelements[ 11 ]. DEF Figure2-10. Stepsofadvancingfrontmethodaredepicted. 36

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11 17 73 77 78 81 ]aresuitableformostoftheengineeringapplications,sincetheideaisrelativelyefcientandhasaneasyimplementationcomparedtoDelaunayandquadtreebasedmethods.Inaddition,inputboundarycanbewell-representedandtheoutputmeshesusuallycontaintriangleswithorientationswhichareappropriateforparticularengineeringproblems,suchasproblemsrelatedtouiddynamics[ 11 ].However,ingeneraladvancingfrontmethodsdonotguaranteeneithersizenorqualitybounds.Asanexceptiontothis,theDelaunayrenementalgorithmgivenin[ 118 ]canbeseenastherstadvancingfronttypeoftechniquethatprovidestheoreticalguarantees.Basicideaistoincrementallycreatenewelementsbeginningfromtheinputboundaryandadjustingtheelementsalongtheway.Ineachstep,anappropriateplaceisfoundforaSteinerpointwhilekeepingandupdatingthefront.However,determiningthepointlocationsisadifculttask.Researcherstrieddifferentstrategies,suchasputtingpointsonthepre-denedcontourlinesorusingwell-knowncurvesforthepointlocations[ 11 ].AnillustrationofadvancingfronttechniqueisshowninFigure 2-10 .Inaddition,Figure 3-9 givesacomparisonbetweenaforementionedtriangulationmethods.Notethat,quadtree-basedmethodinsertssignicantlylargernumberofpointsandgenerateaxis-orientedtriangulations. ABC Figure2-11. Outputofdifferentrenementalgorithms.A)DelaunayRenement.B)QuadtreeRenement.C)AdvancingFront. 37

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15 49 87 107 ].Severalqualityconstraintsaimedtobesatisedusingcirclepackings.Forexample,Bernetal.[ 15 ]andEppstein[ 49 ]gavealgorithmsgeneratingnon-obtusetriangulations,wherealltheanglesarenolargerthan=2.Moreover,ShimadaandGossard[ 107 ]introducedbubblemeshing,whichusestherelationshipbetweenVoronoidiagramandtightlypackedcirclestogeneratequalitytriangulations.Milleretal.[ 87 ]alsoemployedcirclepackingstogetherwithquadtreesandDelaunaytriangulationstoprovidequalitytriangulationsfornumericalmethods.Figure 2-12 givesanexamplecirclepackinganditstriangulation. AB Figure2-12. Circle-basedtriangulationexample.A)Circlepacking.B)Outputtriangulation. 38

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11 ].Thiscanbealsoincorporatedwithpotentialedgeippingoperationstoobtainhigherqualitytriangulations.Theideaistoperformlocalrelocationsandimprovetheoverallshapeorsizequalityofatriangulation.Therelocationstrategycanbebasedonanenergyfunctionorevenaheuristic.Therearemanysmoothingmethodswhicharerelyingondifferentapproaches,suchasgeometry,statisticsandoptimization[ 2 3 8 16 19 20 24 39 46 56 59 91 92 120 122 123 125 ]. 56 ](seeFigure 2-13 ).Thistechniqueiswidelyusedduetoitssimplicityandeffectiveness.However,itdoesnotguaranteequalityimprovementwhereinvertedelementsmightbegenerated.Therefore,manystudiesintroduceddifferentversionsofLaplaciansmoothingorcombinedwithothermethods.Anoverviewofthosecanbefoundin[ 20 ]. AB Figure2-13. Relocatingapointtothecentroidofitsneighbors.A)BeforeLaplaciansmoothing.B)AfterapplyingLaplaciansmoothing. OneofthevariationsofthestandardLaplaciansmoothingalgorithmisthesmartLaplaciansmoothing[ 20 59 ]whichisalsocalledconstrainedLaplaciansmoothing.Themaindifferencebetweentheoriginalandthisvariantisrelocatingpointsonlyifthere 39

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37 ].Thisspecialstructurehasbeenshowntohaveapplicationsinmanyelds,suchasimageprocessing,clustering,celldivision,andothers.Duetal.discussedmanypropertiesandapplicationsofCVTin[ 37 39 122 ].AnimmediatealgorithmforgeneratingsuchtessellationsistoiterativelycomputetheVoronoiregionsforeachpointandupdatingtheirlocationstothecentroidofeachregionuntilconvergence.AdiscussiononotherdeterministicandprobabilisticmethodsforCVTgenerationcanbefoundin[ 37 ].Theiterativemethodisnaturallysuitableformeshsmoothing.Thus,researchersproposeddifferentsmoothingtechniquesbasedoncentroidalVoronoitessellationconcept[ 2 24 39 122 ].ChenandXu[ 25 ]studiedDelaunaytriangulationsintermsoflinearinterpolationerrorforagivenfunctionandintroducedoptimalDelaunaytriangulation(ODT),whichisstronglyrelatedtoCVTs.ThishelpedChentodesignanewmeshsmoothingstrategy[ 24 ],ODT-basedsmoothing,whichaimstoequallydistributetheedgelengthsbasedonthefunctiontobeapproximated. 125 ]proposedasmoothingmethodwhichaimstoimprovethegeometricqualityofameshinacomputationallyeasyway.TheirmethodhasbeenshowntoprovidebetterresultsthanLaplaciansmoothinginqualityandtoavoidcreatinginvertedelements.Inthismethod,anglesinalocalneighborhoodisapproximatelydistributedinequaltoachievebetterqualityelements. 35 60 120 ].Therefore,inordertobenetfrommesh 40

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59 91 ]. AB CD Figure2-14. Applyingdifferentsmoothingmethodsonagiveninputmesh.A)Initialmesh.B)Laplaciansmoothingoutput.C)CVT-basedsmoothingoutput.D)Angle-basedsmoothingoutput. Figure 2-14 demonstratesdifferentsmoothingmethods,whereeachtechniqueprovidesabetterqualitymeshthantheinputtriangulation.Smoothingisconsideredasaseparateoperationwhichisoutsideofthemesh(triangulation)generationprocess.However,fewresearchersalsostudiedintegrationofrelocationandrenementindifferentways[ 32 53 117 ]. 41

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6 105 ].Ingeneral,thebettertheshapeofthetriangles,thesmallertheinterpolationandapproximationerrorsareintheiruse.Atriangleisusuallyconsideredtohaveagoodshapeifitssmallestangleisboundedfrombelow(byauser-speciedconstraintangle),whichimpliesanupperboundonitslargestangleaswell(2).Therearealsoapplicationswhichrequiresnon-triviallysmalllargeangles,wherealgorithmsfocusingonlargeangleswhilegeneratingtriangulationsareneeded.Unlessnewverticesareaddedintothedomain,evenusingDelaunaytriangulations,whichareoptimuminmaximizingthesmallestangle[ 43 ],mayresultinoutputthatinvolvebadqualitytriangles.Hence,qualitySteinertriangulationsareemployedforefciencyandeffectivenessinapplications.However,itisimportanttokeepthetriangulationsizesmallwhilehavingtheshapeofitstrianglesgood,inparticulartheangles.Here,wegeneralizethequalitySteinertriangulationproblemwithangleconstraintsasfollows: 1 : 1 ,where=2. 1 ,where=0. 1 ,where==2. 1 ,where==2(isasmallconstant).Combiningtheobjectivesofthesewell-studiedproblemsgivesusamorecomplicatedproblemthaneitherofthem.Forinstance,itisclearthatDelaunaytriangulation(i.e., 42

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1 2.3.1.1DelaunayrenementSinceDelaunaytriangulationsareknowntomaximizethesmallestangle,manyemployedthemtogeneratenosmallangletriangulations[ 11 26 28 61 87 99 ].TheDelaunayrenementmethodinvolvesrstcomputinganinitialDelaunaytriangulationoftheinputdomainandtheniterativelyaddingSteinerpointstoimprovethequalityofthetriangulation.VarioustypesofSteinerpointsarestudiedintheliterature,whichwereviewbelow. 28 99 104 ]).SeeFigure 2-15 (A).InsertionofitscircumcentersurelyremovesabadtrianglefromtheDelaunaytriangulation(thankstotheemptycirclepropertyofDelaunaytriangles).Moreover,circumcentersoftrianglesstandreasonablyfarfromtheexistingvertices,apropertywhichserveswellforprovingupperboundsontheoutputsizeofDelaunayrenementalgorithms[ 99 ]. 45 ]suggestedinsertingsinksofbadtriangles,whicharecircumcentersofacutetriangles.Foreachbadtriangle,aniterativewalkinthetriangulation,eachtimecrossingtheedgeoppositetotheuniqueobtuseangle,leadstoitssink.SeeFigure 2-15 (B).Sinksareatthelocalmaximaofthelocalfeaturesizefunction[ 45 99 ].Notethatthesinkofabadtrianglecanbequitefarfromthetriangleitself.Hence,abadtrianglemayremaininthetriangulationevenafteritssinkisinserted.However,itisshownthatiterativeinsertionofsinkseventuallyremovesallbadtrianglesinthetriangulation.Experiments[ 45 ]suggeststhatthesinkinsertion 43

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Figure2-15. TypesofSteinerpoints.A)Circumcenter.B)Sink.C)Off-center. 118 ].Off-centersandcircumcentersdifferforverybadtriangles(thosewithsmallestangleatmost=2)andarethesamefortrianglesthatarealmostgood(thosewiththesmallestanglebetween=2and).Intheformercase,theoff-centerofatrianglewiththeshortestedgepqisapointoonthebisectorofpqfurthestfromp(orq)suchthattheangle\poqis.SeeFigure 2-15 (C).Inpractice,theoff-centerinsertionalgorithmcomputessignicantlysmallertriangulations[ 118 ].Off-centersarealsonumericallymorestablethancircumcentersandfacilitatemorerobustsoftware.Theideaofusingoff-centersalsoledtothedesignofthersttime-optimalDelaunayrenementalgorithm[ 68 ].RecentyearswitnessedaninationinthedevelopmentanduseofDelaunayrenementalgorithms[ 32 45 68 69 84 85 103 104 118 ].Someoftheseresearchstudiesmainlyconcentrateonthetheoreticalbounds[ 68 69 84 ],whileothersemphasizethebenetsonthepracticalside[ 103 104 118 ].Milleretal.proposedatime-efcientDelaunayrenementalgorithm[ 69 84 ].However,theiralgorithmcurrentlylacksexperimentalsupporttoindicateitsrelevanceinpractice.Noneofthese 44

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13 ].However,inpracticequadtreerenedmeshesaresignicantlylargerthanDelaunaymeshes.Moreover,thequadtreeimplementationslacktheexibilityfortheusertospecifyaconstraintangle.Instead,axedangle(exactvalueofwhichdependsonthetypeofinput)guaranteeisprovided.Forpolygonaldomains,itisproventhatthequadtreemethodcancomputeoutputtriangulationswhoseanglesareatleast18.4.Inpractice,itiscommontoobserveanglesthatarearound20inthequadtreerenedtriangulations.ThereareotherrenementstrategieswhichareusedinconjunctionwithDelaunaytriangulations.Forinstance,thelongest-edgepropagationpathmethodsusethemidpointofthelongestedgeofcertaintrianglesastheSteinerpoint[ 98 ].TheoreticalboundsforthesemethodstendtoberelativelyweakerthanthatoftheDelaunayrenementmethods.Moreover,thesemethodsalsosufferfromtheterminationprobleminpracticeforconstraintanglesevensmallerthan34. 1 .Forinstance,therearealgorithmsforcomputinganon-obtusetriangulationofsimplepolygons(usuallywithholes)[ 9 12 13 15 49 ].Sincemostofthesealgorithmsuseaxis-alignedstructures,suchasquadtrees,rectangulargridsorvertical/horizontallinesduringtheirconstructionprocess,congurationoftheoutputtriangulationanditssizearenegativelyaffected[ 9 12 13 ].Distinctly,Bernetal.[ 15 ]andEppstein[ 49 ]utilizenon-overlappingdisksforgeneratingnon-obtusetriangulations.However,theanglesintheoutputcanstillbearbitrarilysmall. 45

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B C Figure2-16. Outputofdifferenttypesofrenementalgorithmsandtheiranglehistograms.A)Circumcenterinsertion,[,]=[30,119],613points.B)Off-centerinsertion,[,]=[30,118],467points.C)Quadtreerenement,[,]=[6,156],2893points. Forplanarstraightlinegraphs,Mitchell[ 90 ]andTan[ 114 ]proposedalgorithmsbasedonpropagatingpathsofSteinerpointstoeliminatebadqualitytriangles.Thesealgorithmsgive157.5and132boundsonlargeangles,respectively.Milleretal.[ 86 ]recentlyintroducedasimpleralgorithmwhichusesDelaunayrenement.However,theirupperboundof170isworsethantheboundsprovidedbythepreviousalgorithmsand 46

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13 22 79 80 ].Intheirseminalpaper,Bernetal.[ 13 ]presentavariantoftheirquadtreerenementalgorithminordertogenerate[=36,=80]-triangulations.However,theinputfortheirmethodislimitedtopointsets.Moreover,outputsizeofthisalgorithmtendstobetoolargebecauseitrequiresextralevelofrenementonthequadtree.Also,somestudiesconsiderednosmallangleconditionwithnon-obtuseness.Bakeretal.[ 7 ]employedaregulargridtoconstructnon-obtusetriangulationswhosesmallestangleisaslargeas13.MelissaratosandSouvaine[ 82 ]improvedthisbyintegratingstrategiesofBakeretal.[ 7 ]andBernetal.[ 13 ]andprovidingoutputsizeguarantee,intheory.However,thesealgorithmtendtobecomplicatedandastheyrelyongridstructure,outputsizeforthesealgorithmsarenotpromisingtobecompetitivelysmallinpractice.Indeed,wearenotawareofanysoftwarebasedonthesemethods. 125 ]specicallyfocusesonangles,whileitsbasicideareliesonimprovingtheangledistributions.In 47

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55 ].Inthisstudy[ 55 ],inconsistentbehaviorsofsmoothingmethodshavebeenobserved,especiallywhentheinputdomaincontainscomplexfeatures.Thatis,outputexamplesofsomesmoothingmethodswithsignicantdecreaseinsmallandlargeangleboundsareillustrated(seealsoFigure 2-17 ). 3 35 ].Optimization-basedsmoothingmethodsaregenerallylessefcientthantherenementmethodsasittakesseveraliterationsfortheoptimizationtoconverge.Combinationwithsimplegeometry-basedapproacheshasbeenalsoconsidered[ 59 91 ].Thesealgorithmsmostlyfocusonminimizingthesmallestangleratherthanthelargestangle,sincethelatterisknowntobeharderthantheformerduetothenon-convexnatureoftheproblem[ 3 ].However,arecentwell-centeredtriangulation(WCT)methodofVanderZeeetal.[ 120 ]containsanoptimizationfunctionwhichaimstocomputeacutetriangulations(seeFigure 2-17 ).ExperimentsindicatethatWCTmethodgenerallyimprovesthelargestanglebutdoesnotnecessarilyproduceacuteangletriangulationsandalsosometimesdeterioratesthesmallestangle.Notethat,sincesmoothingmethodsworkonexistingmeshes,theimprovementontheanglequalitystronglyrelatedtothequalityoftheinitialtriangulationandexpectedtobelimited. 48

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BC DE BCDE Figure2-17. Outputofdifferenttypesofsmoothingalgorithmsandtheiranglehistograms.A)Initialtriangulation,[,]=[0.65,173].B)Laplaciansmoothing,[,]=[13.3,140].C)Angle-basedsmoothing,[,]=[8.3,147].D)CVT-basedsmoothing,[,]=[7.7,147].D)WCTsmoothing,[,]=[2.65,141]. 49

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13 99 ].ApproximationguaranteeforaspecicSteinertriangulationalgorithmisusuallyexpressedasfollows:Givenatwo-dimensionaldomainandaconstraintangle,thealgorithmcomputesatriangulationoftheinputdomainwhosesizeiswithinaconstantfactoroftheoptimalsuchthatalltheanglesofthetriangulationareatleast.Theangleshouldbethoughtofasthetheoreticallimitofanalgorithm.Theexactvalueofthedependsontheapproach/algorithmaswellastheinputtype.Forinstance,theDelaunayrenementalgorithmofRuppert[ 99 ]isproventocomputegoodtriangulationsfor=30onpointsetsandfor=20.7onplanarstraightlinegraphs.Inpractice,thelackoftheoreticalguaranteeforaSteinertriangulationalgorithm,when>,isobservedasanever-endingrenementprocess.(SeeFigure 3-1 (A).)Steinerpointsareiterativelyinserteduntilthecomputerexceedsitsnumericalprecisioncapacityoritsmemory.Shewchuk'sexperimentalstudyrevealedthattheDelaunayrenementalgorithminpracticeworksbetterthanitstheoreticalguarantee: Ruppert[ 99 ]provesthatthisprocedurehaltsforangleconstraintofupto20.7.Inpractice,thealgorithmgenerallyhaltswithanangleconstraintof33.8,butoftenfailstoterminategivenanangleconstraintof33.9.Itwouldbeinterestingtodiscoverwhythecut-offfallsthere.[ 101 ]Therehavebeenattemptstoexplainthiscut-off[ 85 ].Here,wetakeadifferentstrategy,and,ratherthanexplainingthislimitation,weremoveit.WepresentanewDelaunayrenementalgorithmwhichsetsanewcut-off.Thiscut-offanglewillbereferredtoas,representingtheempiricallimitforDelaunayrenementalgorithms.Ourrenementalgorithmterminateswithgoodtriangulationsforconstraintanglesupto42,i.e.,=42.Figure 3-1 illustratestheoutputofouralgorithmincontrastwiththebehaviorofpreviousDelaunayrenementalgorithms. 50

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B Figure3-1. TheoriginalDelaunayrenementalgorithmdoesnotterminateforconstraintangleslargerthan34.Ouralgorithmcomputesqualitytriangulationsforconstraintanglesupto42.Forexample,whentheconstraintangleis40,theoriginalDelaunayrenementchokeswithbad(shaded/red)triangles(A),whereasouralgorithmcomputesanicelygradedtriangulationwhereallanglesarelargerthanorequaltotheconstraintangle(B). 51

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3.2 ,weclassifythisSteinerpointdescriptionintofourdifferenttypes.Oursecondalgorithmincorporatesasimplerelocationschemewhichbecomeseffectiveespeciallyforlargevalues.BeforeattemptingtoinsertaSteinerpointforabadtriangle,wesimplytryxingthebadtrianglebyrelocatingoneofitsvertices(ifthatvertexwasinsertedasaSteinerpoint).TheseideasleadtothefollowingasthemaincontributionsofthisChapter,whichweelaboratemoreviaexperiments: 52

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3-2 ).Inpractice,suchaphenomenonstartsoccurringfor>34,regardlessofthetypeanddistributionofinputdata.SeeFigures 3-3 and 3-4 .Coincidentally,allknownpreviousversionsoftheDelaunayrenement[ 28 68 84 99 102 118 ]havetheterminationproblemaround34,asfurtherexplainedinthenextsection. When>30,circumcenterinsertionintroducesshorterpairwisedistances(amongverticesoftriangulation)thanexistingones,i.e.,jpcj=jqcj=jrcj
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3-3 illustrateshowbothTriangle1.4,whichisanimplementationofthecircumcenterinsertion,andTriangle1.6,whichisanimplementationoftheoff-centerinsertion,sufferfromtheterminationproblem.Itisimportanttoobservethecharacteristicdifferencebetweenthesetwomethodswhentheygetcaughtinthenever-endingrenementprocess.Thisdifferenceisduetotheorderinwhichthebadtrianglesarehandled.Theoff-centerinsertionalgorithmworksbetterwhenthebadtriangleswiththeshortestedgesarehandledrst[ 68 ].Ontheotherhand,thecircumcenterinsertionalgorithmworksbetterwhenthebadtriangleswiththesmallestanglesarehandledrst.Whentheyterminate,theoff-centerinsertionperformsbetterthanthecircumcenterinsertion;thatis,itoutputsmesheswithfewertriangles[ 101 ].However,wheninterruptedinthenever-endingrenementprocess,theevolvingmeshoftheoff-centerrenementlooksworsethanthatofthecircumcenterrenement.Thiscanbexedbychangingthebadtrianglehandlingordertosmallestlengthrst. 85 94 103 ].Itisimportanttonotethatingeneraltheterminationproblemexistsregardlessofthesimplicityoftheinputdomain.SeeFigure 3-4 ,wheretheinputconsistsofapairofpointsenclosedbyalargehexagon.Ingeneral,foranyinputdomain,itiscommontoobserveoverrenementoccurringinregionsfarawayfromtheinputfeatureswheneverislarge.Inthisstudy,wedonotlimitourselvestocertaintypesofconstraintsandratheraddresstheterminationproblemingeneral.OuranalysisinSection 3.3 focusesonpointsets.However,thisworkcanbeeasilycoupledwiththepreviouswork[ 85 104 118 ]thatconsideredplanarstraightlinegraphs.Ourexperiments,ondatasetsthatareplanarstraightlinegraphswithsmallinputangles,verifythisstatement. 54

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CD EF Figure3-3. Terminationproblemshownfortheconstraintangle=36ontherabbitdataset.Badtrianglesareshaded(red).LeftA,C,andE:evolvingtriangulationusingthecircumcenterinsertion(Triangle1.4)using100,500and5000Steinerpoints,respectively.RightB,D,andF:evolvingtriangulationusingtheoff-centerinsertion(Triangle1.6)using100,250and500Steinerpoints,respectively.Neitherofthetwoalgorithmsterminates.So,forthisillustration,weinterruptedtheirexecutionaftertheinsertionofSSteinerpoints. 55

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Figure3-4. Terminationproblemisshownfor=35onasimpledataset:apairofpointsatunitdistancefromeachotherinsideahexagonofsidelength12units.Badtrianglesareshaded(red).A)Delaunaytriangulationoftheinputwhichincludestwobadtriangles.BandC)Evolvingtriangulationusingthecircumcenterinsertion(Triangle1.4).Therenementprocessdoesnotterminate,andhencetheexecutionisinterruptedaftertheinsertionof0,50and1000Steinerpoints. 109 ]recentlyshowedempiricalevidencethatstandardDelaunayrenedmeshes(throughcircumcenterinsertion)areroughlytwicethesizeoftheoptimalmeshesforanapplicationhecallsfunctionapproximation.OurresultsherecomplementthisstudyaswegetasizeimprovementofaroundafactoroftwocomparedtothepreviousDelaunayrenementalgorithms. Denition12.

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68 ]provedthatthetriangulationofapointsetisgoodifandonlyifthepetalofeverypairofpointscontainsanotherpoint.Inthefollowingalgorithm,wesuggestonetopickaSteinerpointinsideemptypetalsfurthestawayfromallexistingvertices;thisistheconceptoflocallyoptimalpointlocation. ComputetheDelaunaytriangulationoftheinput ThispointiseitheraVoronoivertex(butnotnecessarilythecircumcenterofpqr)oronaVoronoiedge(seeLemma 1 ).Weclassifythesepointsintofourdifferenttypesinordertorelatethemwiththeexistingrenementstrategies.Thisclassicationalsohelpsustopresentatheoreticalandexperimentalassessmentofourmethod.Letpqrbeabadtrianglewhoseshortestedgeispq.Then,thepointinsidethepetalofpqfurthestawayfromallexistingverticesisoneofthefollowingtypes(seeFigure 3-5 ). 118 ]. 3-5 (C)).Hence,ouruseofsinksissomewhatdifferentthantheoriginalsinkinsertionalgorithmofEdelsbrunnerandGuoy[ 45 ]. 57

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CD Figure3-5. Findapointxinsidethe(shadeddisk)petalofpqthatisfurthestawayfromallexistingvertices.SuchapointcanbeonaVoronoiedge(AandB)oraVoronoivertex(CandD).A)TypeI:onthedualofpq.B)TypeII:onaVoronoiedgeotherthanthedualofpq.C)TypeIII:acircumcenterotherthanthatofpqr.D)TypeIV:thecircumcenterofpqr. NotethatthefurthestpointinanarbitraryregionfromallexistingverticesisnotnecessarilyontheVoronoiskeleton.AstraightforwardexampleofthisisaclosedregionwhichdoesnotcontainanyVoronoiedgeorVoronoivertex.Alocallyoptimalpointlimitedtosucharegionisclearlyonitsboundary.Ontheotherhand,Figure 3-6 (A)illustratesamoreinterestingexamplewheretheregionofinterest(shaded)intersectstheVoronoiskeleton.Notethattheregioninthisexampleisconvexandalsocontainstwoverticesonitsboundary,twoofthepropertiesofapetal.ThelocallyoptimalSteiner 58

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1 AB Figure3-6. A)LocallyoptimalSteinerpointlimitedtoaconvexregion.B)IllustrationoftheLemma 1 proof. 3-6 (B).)Sincejp0qj
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3-6 (B)).Observethateitherporqisclosertos0thansis.So,s0=2Voronoi(s),andhences06=x.Alsonotethat,foranypointotherthans0on@petal(pq),itispossibletondanotherpointfurtherfromsbywalkingon@petal(pq)towardss0.Thisinturnleadstoacontradictionasxcannotbeoptimalunlessitisbothon@petal(pq)andontheVoronoiskeleton. Wecomputethelocation(andalsothetype)oftheSteinerpointsimplybydoingalocalsearch(breadth-rstsearch)ontheVoronoigraph.Theanglesofthevisitedtrianglesprovideguidanceinthissearch.Forinstance,ifthesmallestangleofthebadtriangleislessthan=2,thenweknowforsurethatthelocallyoptimalSteinerpointisofTypeI(anoff-center).Ingeneral,itisbettertostartthesearchvisitingthetriangleoppositetothelargestangleofthebadtriangle.WeelaboratemoreontheimplementationdetailsofthealgorithminSection 3.5 118 ]forananalysisoftherenementthatusesTypeIandTypeIVSteinerpoints.Indeed,onemightexpectanimprovementontheanglebound.Suchanimprovement,however,seemsdifculttoproveandisleftasanopenproblem.Here,weprovidesometheoreticalevidenceonwhyourstrategyworksforlargevaluesofconstraintangle.Forthispurpose,thefollowinglemmashouldbecomplementedwiththeexperimentalresultsonthepercentageuseofeachtypeofSteinerpoint(presentedonTables 3-2 and 3-3 inSection 3.5 ). 60

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Figure3-7. A)TypeII(A).B)TypeII(B). WefurtherclassifytheTypeIIverticesforthesakeoftheanalysis.TwospecialcasesofTypeIIwillbeanalyzedseparately,wheretheSteinerpointxisonthedualoflongestedgeofthebadtriangle.Inotherwords,consideraSteinerpointthatisonthedualedgeofqrofabadtrianglewithshortestedgepq,whereqristhelongestedge.ThisTypeIISteinerpointiscalledaTypeII(A)if\qpr=2andaTypeII(B)if\pqr\qpr<=2. 2 doesnotcreateanyfeatureshorterthantheexistingoneswhileinsertingavertexxof (a) TypeISteinerpointsif=3=60; (b) TypeII(A)Steinerpointsif=4=45; (c) TypeII(B)Steinerpointsif=5=36; (d) TypeII,TypeIII,andTypeIVSteinerpointsif=6=30.Proof.Letpqrbeabadtrianglewithshortestedgepq.(a)SincexisontheVoronoiedgeofpq,itsnearestneighborsamongtheexistingverticesarepandq.Observingthatjxpj=3completesthispartoftheproof.(b)Assume,withoutlossofgenerality,thatxisontheVoronoidualofqr.So,\qprisanon-acuteangle.Letlbethelineorthogonaltopqwhichgoesthroughp

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3-7 (A)).Letybetheother(thanp)intersectionoflineland@petal(pq).Withoutlossofgenerality,assumethatpetal(pq)isunitdisk.Then,thelengthofthearcpqis2(inradians).Letx0bethemidpointofthearcyq.Thismeansthelengthofthearcx0qis=2.Hence,=4ifandonlyifpqx0q.Notethatthereisnopointbelowthelinelandoutsidethepetal(pq)thatisclosertox0thanq.So,jx0qjjxqj.Then,weconcludethatjxqjjpqjif=4.(c)Thispartoftheproofissimilarto(b).Assume,withoutlossofgenerality,thatxisontheVoronoidualofqr.So,\pqr\qpr<=2.Letlbethelineorthogonaltopqandgoingthroughthemidpointofpq(Figure 3-7 (B)).Letybetheintersectionpointoflineland@petal(pq)thatisfurthestfrompq.Withoutlossofgenerality,assumethatpetal(pq)isunitdisk.Then,thelengthofthearcpqis2(inradians)andthelengthofthearcyqis.Letx0bethemidpointofthearcyq.Thismeansthelengthofthearcx0qis=2=2.Hence,=5ifandonlyifpqx0q.Notethatthereisnopointbelowthelinelandoutsidethepetal(pq)thatisclosertox0thanq.So,jx0qjjxqj.Then,weconcludethatjxqjjpqjif=5.(d)Theproofisstraightforwardtoshow. Lemma 2 suggeststhatifwecouldshowthattheAlgorithm 2 onlyuses,say,TypeIandTypeII(A)Steinerpoints,thenitwould(provably)computetriangulationswithaminimumangleof45.Unfortunately,suchapremisedoesnotseemplausible.Ourexperiments(presentedinSection 3.5 )showthatallfourtypesareemployedbyAlgorithm 2 indifferentamounts.Thefollowingtheoremfollowsfromtheabovelemma.ItsproofissimilartotheresultsintraditionalDelaunayrenement[ 99 104 118 ]andhenceomittedhere. 2 asisthecaseformostotherDelaunayrenement 62

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68 ].ThistechniqueemploysabalancedquadtreeasadatastructureandtakesadvantageofthelocalityoftheSteinerpointswithrespecttotheshortestedgeofbadtriangles. Figure3-8. Relocatingafreevertexofabadtrianglepqrtotheintersectionofthepetalsofthelinkofa=r. 3.3 )andexperimentswithAlgorithm 2 ,weobservedthattherenementprocessintroducesshorteredgesthanexistingonesusuallywhenabadtriangleisalmostgoodandalsotheneighbortrianglesaregoodoralmostgood.Thissuggeststhatitmightbeeasytoxsuchbadtrianglesbyalocalsmoothingstrategy(seeFigure 3-8 ).Whileonemightexplorevariouspowerfulsmoothingstrategies[ 3 35 60 ]inthesecases,weoptforsimplicityandefciency.Werstrecallacoupleofdenitions,someofwhicharesurveyedin[ 44 ],andthendescribeasimpleadjustmenttoouralgorithm.Thestarofavertexaconsistsofalltrianglesthatcontaina(seeFigure 3-8 ).Thelinkofathenconsistsofalledgesoftrianglesinthestarthataredisjointfroma.AvertexissaidtobefreeifitwasinsertedbytherenementalgorithmasaSteinerpoint.Inputverticesarenotfreeandneverrelocated.Foreachbadtriangle, 63

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2 ComputetheDelaunaytriangulation(DelTri)oftheinput LetPdenotethemaintainedpointset ifrelocated==FALSEthen 3 duetothepointrelocationstep.Whileprovinganimprovedangleboundforitisleftopen,wecanmatchtheearlierboundsbyasimplemodicationtoAlgorithm 3 :Applytherelocationonlywhentheconstraintangleislargerthan30.ThismodicationkeepsAlgorithm 3 stilleffectiveforlargeconstraintanglesandprovablygoodforsmallconstraintangles. 3-1 3-9 3-10 3-11 3-15 3-16 ,and 3-17 .)However,theperformanceplotsinFigures 3-12 3-13 3-14 3-18 ,and 3-19 andthestatisticsinTables 3-1 and 64

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revealmuchmoreinformationregardingtheperformanceofouralgorithmsincomparisontobestknownpreviousmethods.Furthermore,Table 3-3 showsthebehaviorofAlgorithm 3 intermsofthetypesofSteinerpointsused. 83 ].Ingeneral,foraDelaunayrenementalgorithm,insertingaSteinerpointxofabadtrianglepqr(withtheshortestedgepq),doesnotguaranteethatpqxisagoodtriangle.Forinstance,inthecircumcenterinsertionalgorithmofRuppert[ 99 ],thiscouldhappenbecause\pxqcanbestrictlysmallerthan.Whereas,inouralgorithmforTypeIIandTypeIIISteinerpoints\pqxor\qpxcanbesmallerthan.Inourimplementation,whenever\pqxor\qpxislessthan,weoptforusingTypeIVSteinerpointtomakesurethatpqxisagoodtriangle.IntherelocationstepofAlgorithm 3 ,weavoidcomputinganexactintersectionofthesetofpetalsonthelink.Instead,weusedanapproximaterepresentationofthefeasibleregionofeachpetal,wherethetwoconstraintsimposedbytheanglesincidenttotwolinkverticesarealsoconsidered.Then,computeanapproximatefeasibleregionsimplybyahalf-planeintersectionalgorithm.Ourexperimentsindicatethatusingan8vertexconvexpolygonforapproximatingeachsuchregionisefcientandeffective.Westartthecalculationsfromthosethatarefurthestfromeachotheronthelink.Ifthereexistsanypetalpairwithnointersection,thenweterminatetheprocessandconcludethatnorelocationpointisfound.Otherwise,wecomputethecentroidoftheintersectionregionbsuchthatalltrianglesofstar(b)intheDelaunaytriangulationoftheset(P[fbgfag)areofgoodquality,wherePistheevolvingpointsetandaisthe 65

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3-8 ).WealwaysmaintaintheDelaunaypropertyofthetriangulation.Alternativestrategiesforpointrelocationcanbeexplored.Wechoosethisparticularoneforitssimplicityandefciency. AB Figure3-9. OutputoftheAlgorithm 3 forBoeingandoctagondatasetsforlargevalues(=42).Thepracticallimitofouralgorithmforthesedatasetsareactuallyhigherthan42asinTable 3-1 3-1 66

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PerformanceofAlgorithm 3 withrespecttoTriangle1.4(Ruppert'salgorithm)isshown.Asadditionalinformation,averagevaluesforTriangle1.6andAlgorithm 2 are35.08and38.40,respectively.(Continuedonthenextpage.) Cut-offCut-offSizeangle()angle()improvementDataset#Points#Segments#HolesTriangle1.4Algorithm 3 (=30) Cyprus448447033.942.32.82Malta162162034.442.42.57Tri.&Tobago429429034.042.42.78Turkey216216034.142.42.51PuertoRico7777034.842.42.56Hawaii13341334034.042.22.96Italy10791079033.942.32.72Japan77017701033.742.22.78Florida304304033.942.42.81California291291033.942.42.68Texas641641033.842.32.83NewYork282282034.042.42.72Superior522522734.242.41.99Fish9292135.042.52.10Kangaroo112112135.542.41.37Koala163163434.542.41.81Rabbit186186334.442.41.99Cheetah304304434.442.41.84Buttery3793791434.242.32.36Bat194194334.842.41.88Dolphin200200134.342.42.37Goose234234234.542.42.39Rhea230230134.342.42.00Goldsh26852685133.842.32.58Crab31683168233.842.32.20Giraffe14531453234.242.42.43Alligator11001100334.042.32.94Blackcat236236134.342.42.26Cat848848134.242.42.46Hexagon86034.842.71.82Octagon108036.342.63.20Boeing3030134.042.44.14Boxedpair64034.242.63.25Kuzmin20000033.742.33.19Plate6564534.542.41.58Random10000033.842.42.66 67

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3-1 :Continued. Cut-offCut-offSizeangle()angle()improvementDataset#Points#Segments#HolesTriangle1.4Algorithm 3 (=30) IshakPashac.165016502333.842.22.72Ayasoa13741374034.042.42.44KzKulesi754754034.142.42.64NewYorkview12411241034.142.42.40TajMahal379379034.542.42.43Median304291N/A34.242.42.51Average844746N/A34.342.42.49 NumberofdifferenttypesofSteinerpointsusedbyfourdifferentDelaunayrenementalgorithms:Circumcenterinsertion,off-centerinsertion,Algorithm 2 andAlgorithm 3 .Datasetlegend:K,Kuzmin;C,California;J,Japan;I,IshakPashacastle;D,Dolphin. DataCCtrOff-centerAlgorithm 2 Algorithm 3 setTypeIVTypeITypeIVTypeITypeIITypeIIITypeIVTypeITypeIITypeIIITypeIVRel. K20189538489032832344233273093483115K307938131026901009881100441101274068633427K341244355351470144515561251456123697980639K37111211724212498313194918581248190944K4111111113648409126157192057C2026175786345390633428022C3079217128611912089112011153337C341298532163216120717018079351C3711125537923014216299108495C41111111139357319839160J205049119621701033986952710658597019378J3018493353261482479287119536025502404133834971J341646211793350046262972157347339051887971345J371114956730247584684680576526232211881J411111111985914178714316985834I20159644774737830041124022492934178I30496111111815760785509147556973592240I34118353327963129978842997108652519355I371111331196013171251299160875552479I4111111112392330615193901168D2012032543112330311126012D30329761365547533573941227D3413121052496774824686556134D37111881109811899456637D41111111114421212428103 68

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B Figure3-10. OutputoftheAlgorithm 3 fortwocomplexdatasets,Italyandbuttery,forlargevalues(=41and=41.5).Thepracticallimitofouralgorithmforthesedatasetsareactuallyhigherthan41and41.5,respectivelyasinTable 3-1 69

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C Figure3-11. OutputoftheAlgorithm 3 forthreecomplexdatasets,cookinglady,seahorseandChina,forlargevalues(=41).Thepracticallimitofouralgorithmforthesedatasetsareactuallyhigherthan41. 70

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PercentagevaluesofdifferenttypesofSteinerpointsusedbyAlgorithm 3 .Datasetlegend:K,Kuzmin;C,California;J,Japan;I,IshakPashacastle;D,Dolphin. Algorithm 3 DatasetTypeITypeII(A)TypeII(B)TypeII(Rest)TypeIIITypeIV K2033.126.24.01.135.30.3K3041.022.45.81.827.81.3K3438.825.46.51.126.12.1K3737.227.96.80.823.83.6K4132.930.66.20.123.66.5C2050.425.60.80.822.40.0C3041.833.44.21.018.51.0C3439.436.34.21.218.30.7C3734.441.95.60.217.20.6C4132.743.64.00.016.53.2J2040.430.91.00.826.60.3J3040.333.62.81.621.20.5J3437.136.24.11.420.21.0J3735.237.45.10.919.71.7J4130.037.06.00.121.75.2I2042.425.10.90.230.90.4I3041.636.91.40.119.80.1I3438.039.02.10.320.00.7I3735.039.23.70.420.31.4I4131.438.54.90.120.05.1D2045.613.21.51.538.20.0D3041.025.91.40.729.51.4D3435.830.53.20.529.50.5D3736.333.14.50.822.92.4D4128.336.84.90.024.45.5 3-12 3-13 ,and 3-14 aswellasthefthcolumnofTable 3-1 showthattheoriginalDelaunayrenementalgorithmisimpotentforconstraintangleslargerthanthecut-offangleforthismethodwhichis34.Theoff-centerinsertionalgorithmofUngorhasalreadyextendedto35.Ontheotherhand,ourAlgorithm 2 terminateswithcorrectoutputforconstraintanglesupto38.5.Finally,ourAlgorithm 3 worksforconstraintanglesupto42.NotethatplotsinFigures 3-12 3-13 ,and 3-14 arecomposedoftwodifferentparts,mainlydemonstrating.TheupperpartplotsthenumberofSteinerpointsusedby 71

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3 .Allplotsinbothpartsarewithrespecttothesamex-axis,thatis,theconstraintangle.Ourexperimentsaredoneforevery0.1,0.1,and0.5anglesforthegivenrangeinFigures 3-12 3-13 ,and 3-14 ,respectively.Plotsintheupperpartsclearlyshowthecut-offangles,values,foreachofthefouralgorithms.Tables 3-2 and 3-3 listthedistributionofeachtypeofSteinerpointusedbythealgorithmsandmighthelpexplaintheimprovementsachievedbyouralgorithms.Statisticsrevealthat,amongthefourtypeofSteinerpoints,circumcentersareleastfrequentlyusedbyouralgorithms.Therearecaseswherecircumcentersarealtogetheravoided.Ingeneral,eachoftheothertypesconstitutearound30to35%ofthetotalSteinerpoints,onlyatmost7%ofthemarecircumcenters.Asweincreasetheminimumangleconstraint,weobservethatcircumcentersarechosenmorefrequently.Inaddition,Lemma 2 andTable 3-3 togetherfurtherclarifyourresults.ItisalsointerestingtonotethatthenumberofpointrelocationsperformedisquitesmallcomparedtothetotalnumberofSteinerpoints.Itseemsthatonlyaround5%oftheSteinerpointsarerelocatedwhentheconstraintangleissmall(20orless).Thispercentagereachesaround15%forlargerconstraintangles.SeethelastcolumnofTable 3-2 3-12 3-13 ,and 3-14 reectonthenumberoftrianglesintheoutput.Weobservedthatthetwoalgorithmsproposedheregivesignicantoutputsizeimprovementsoverthepreviousalgorithms.Theseimprovementsareparticularlyimpressivewhentheconstraintangleislarge.InFigures 3-12 and 3-13 ,wehighlightthesizeimprovementsforconstraintangles30andlarger,whereasFigure 3-13 presentstheplotsfortheentirerangeofconstraintangles.Sizeratioplots,thebottompartsofthegures,showthebehaviorofthisimprovementwhiletheconstraintanglechanges.(SeeFigure 3-15 3-16 ,and 3-17 also.) 72

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Outputsizecomparisonplotsfortwodatasets(Cyprusandcrab),whichalsorevealthepracticalangleboundsofthefouralgorithms,demonstrating. 73

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Outputsizecomparisonplotsfortwodatasets(hexagonandNewYorkview),whichalsorevealthepracticalangleboundsofthefouralgorithms,demonstrating. 74

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Outputsizecomparisonplotsfortwodatasets(Boeing,Florida),whichalsorevealthepracticalangleboundsofthefouralgorithms,demonstrating.Triangle1.4insertsadditionalpointsontheboundary,evenifnopointinsertionisnecessaryforsatisfyingtheconstraintangle.Largesizeimprovementobservedforsmallvaluesisasimpleoutcomeofthisdifference. 75

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3-1 indicatestheratioofthenumberofSteinerpointsinsertedbyTriangle1.4overAlgorithm 3 for=30.Onaverage,thesizeofthetriangulationsdecreasedaroundafactorof2.Inaddition,Figure 3-18 representsamoreextensiveversionofthisexperimentwhichhasbeencompletedbyusingaround40datasets,constructingqualitytriangulationsofthose,forevery1angleinbetween5and35byusingTriangle1.4,Triangle1.6,andAlgorithm 3 .Basedonthisplot,theoutputsizeimprovementsofouralgorithmincomparisontobothpreviousalgorithmscanbeclassiedassignicant,good,andimpressiveintheconstraintangleranges[520),[2030),and[3035],respectively.Thisplotreveals,forinstance,thatouralgorithmgivesaround50%smallermeshes(averageover40datasets)thanthebestknownpreviousDelaunayrenementimplementationwhentheconstraintangleis30. 2 and 3 runfasterthanthepreviousalgorithms.Betweenthetwoofouralgorithms,Algorithm 3 isslightlyfasterthanAlgorithm 2 .(SeeFigure 3-19 .) 3.5 thatthesizeofthetrianglesinagradedmeshisdictatedbythedomaingeometrydescriptiontogetherwiththeconstraintangle.Formostapplications,furtherconstraintsneedtobeimposedinordertocontaintheerrorsinthesimulationatareasonablelevel. 76

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Figure3-15. OutputsizecomparisonontheBoeingdataset.A)Circumcentermethod(Triangle1.4).B)off-centermethod(Triangle1.6).C)Algorithm 3 .For=30,thenewalgorithminserts57Steinerpoints,almosthalfasmanyasthe118Steinerpointsinsertedbytheoff-centeralgorithmwhichisinturnhalfasmanyasthe236Steinerpointsusedbythecircumcenterinsertionalgorithm.TheoutputoftheAlgorithm 2 looksverysimilartothatofAlgorithm 3 ,with71Steinerpoints. Figure3-16. OutputsizecomparisonontheAlligatordataset.A)Circumcentermethod(Triangle1.4).B)off-centermethod(Triangle1.6).C)Algorithm 3 .For=30,thenewalgorithminsertsmuchfewerSteinerpoints(1376)thantheoff-centerandthecircumcenterinsertionalgorithms(2163and4976respectively).TheoutputoftheAlgorithm 2 looksverysimilartothatofAlgorithm 3 ,with1564Steinerpoints. 77

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Figure3-17. OutputsizecomparisonontheFloridadataset.A)Circumcentermethod(Triangle1.4).B)off-centermethod(Triangle1.6).C)Algorithm 3 .For=32,thenewalgorithminsertsmuchfewerSteinerpoints(346)thantheoff-centerandthecircumcenterinsertionalgorithms(606and1391respectively).TheoutputoftheAlgorithm 2 looksverysimilartothatofAlgorithm 3 ,with392Steinerpoints. Figure3-18. PlotofaverageoutputsizeratioofTriangle1.4andTriangle1.6withAlgorithm 3 intermsofthenumberofSteinerpointsvs.theconstraintangleforalldatasetsshowninTable 3-1 78

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Therunningtimesoffourdifferentalgorithmsareplotted(withrespecttovaryingconstraintangle)forTurkeydataset. Acommonlyusedmeshelementsizeconstraintisthemaximumareaconstraintwhichoftenresultsinuniformmeshes.Trianglesoftware,forinstance,allowstheusertospecifythemaximumareaconstraint.Unfortunately,theadditionalconstraintdoesnothelpTriangletoovercometheterminationproblem.Asisthecaseingradedmeshing,thesoftwaregetsintoaninniteloopwhentheconstraintangleislargerthan34.Below,wesummarizeourresultsontheperformanceoftheproposedalgorithmswhentheyareusedforcomputinguniformmeshes. 3-20 3-21 3-22 and 3-23 ,Algorithms 2 and 3 generallyterminateforconstraintanglesaround38.5and42,respectively,evenwhenamaximumareaconstraintisimposed.ThesevaluesareconsistentwithourpreviousresultsoverdatasetsgiveninTable 3-1 79

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3-24 illustratesthatAlgorithm 3 createsamuchsparsertriangulationthanthepreviousalgorithmsunderthesameminimumangleandmaximumareaconstraints.Itseemsthatweachievedoutputsizeimprovement(foruniformtriangulation)bycreatingtriangleswhoseareasareusuallyquiteclosetothespeciedareaconstraint. AB CD Figure3-20. OutputofAlgorithm 3 forvariousdatasetsforvariouslargevaluesforaxedmaximumareavalue.A)=41,maxarea=0.002,shortestedge=0.0036.B)=41.5,maxarea=0.05,shortestedge=1.C)=41,maxarea=0.00006,shortestedge=0.0018.D)=41,maxarea=20,shortestedge=3.12.Theseandmaximumareavaluesshouldnotbeseenasthelimitofouralgorithmfortheshowndatasets.Valueshavebeenchosenaccordinglytohaveclearexamplegures.Also,theshortestedgelengthinDelaunaytriangulationofthesedatasetsaregivenasreferencesformaximumareavalues.Forabetteridea,seetheplotsinFigures 3-23 and 3-22 80

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Figure3-21. OutputofAlgorithm 3 forTexasdatasetfordifferentmaximumareaconstraints.A)=41.5,maxarea=150,shortestedge=1.B)=41.5,maxarea=325,shortestedge=1. Figure3-22. PlotoftheoutputperformanceratioforthePuertoRicodatasethavingunitdistanceshortestedgelength.TheratiorepresentsthenumberofSteinerpointsinsertedbyTriangle1.4overthecorrespondingnumberofpointsinsertedbyAlgorithm 3 fordifferentconstraintanglesforsomevaryingmaximumareaconstraint.Notethattheratioisalwaysgreaterthan1.15showingthatAlgorithm 3 outperformsTriangle1.4evenwithasmallareaconstraint. 81

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PlotofthenumberofSteinerpointsvs.theconstraintangleforsomexedmaximumareaconstraint.Themaximumareaconstraintvalue(=50)fortheFloridadatasetischosenas50timesthesquareoftheshortestedge(=1.0),whereasfortheTurkeydatasetthemaximumareaconstraint(=130)is10timesthesquareoftheshortestedge(=3.6)ofDelaunaytriangulationsofeachdatasets. 82

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Figure3-24. Outputsizecomparisonontheshdataset.A)Triangle1.4(circumcenter).B)Algorithm 3 .For=20andmaximumarea=18,whichisdoubleofthesquareofshortestedgeinDelaunaytriangulationofthedataset(shortestedge=2.98),thenewalgorithminserts1310Steinerpoints,around17percentlessthanthe1572Steinerpointsinsertedbythecircumcenteralgorithm.Theoff-centeralgorithminserts1584Steinerpoints,whichisalmostthesameasthecircumcenteralgorithm.TheaveragetriangleareasfortheoutputofTriangle1.4andAlgorithm 3 are11.44and13.87,respectively. 3-22 wheretheplotshowsthecomparativebehaviorofAlgorithm 3 andTriangle1.4basedonchangesinthemaximumareaconstraintintermsofoutputsizeratioaswellastheeffectofhavingdifferentconstraintanglevalues.Anumberofevidentobservationsfromthegurecanbesummarizedasfollows:Whenthemaximumareaconstraintisbetween1and10,changinghaslittleeffectbutchangingmaximumareainthisrange,forxed,alsohaslittleeffect(wherethemaximumareaconstraintisdominant).Formaximumareaabove20,changingforaxedmaximumareahasasignicanteffect(wheretheconstraintangleisdominant).Formaximumareaabove100,changingmaximumareaforaxedhaslittleeffect(wheretheconstraintangleisstilldominant).Overall,aswechoosealargerareaconstraint,sizeimprovementoverTriangle1.4increasesmoderatelyandbecomesconstant.Inaddition,increasingtheconstraintangleaffectstheamountofsize 83

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3-23 showstheoverallbehaviorofdifferentversionsofTriangletogetherwithAlgorithms 2 and 3 foraxedvalueofthemaximumareaconstraintoverthecompleterangeofanglesintermsofthenumberofSteinerpointsattheupperpartsoftheplots,whilethesizeratiobetweenTriangle1.4andAlgorithm 3 canbeseenatthebottom.Clearly,comparedtopreviousversionsoftheDelaunayrenementalgorithms,ouralgorithmsproducesmaller-sizedtriangulations.Alsonotethattheratiostayssignicantlyabove1,whichindeedcorrespondstoasizeimprovementaround15%onaverage.Naturally,theimprovementsaremoresignicant,forlargervalues,astheconstraintangleincreases. 3.5.5 ,weappliedsmartLaplacianandangle-basedsmoothingonaninitialqualitytriangulation(=30)forAfricadatasetgeneratedbycircumcenterinsertionstrategy.Ascanbeseen,theanglequalityisslightlyimprovedorevendecreasedaftersmoothingoperation.However,ourmethodgenerateshigherqualitytriangulation(=40)withnearlythesamenumberofpointsinhalfofthetimeittakesforclassicalDelaunayrenement.Inaddition,applyingsmoothingtothistriangulationonlyhelpstosmooththeanglevaluesanddeteriorateordoesnotimprovetheanglequality.Thisindicatesthatourintegratedrelocationandrenementstrategyismorepowerfulandsignicantlyfasterthanthecombinationofgeneratingagoodqualitymeshandapplyingsmoothingafterwards.Itisimportantnottoconfuseouralgorithmwithaglobalmeshoptimizationtechnique[ 3 35 60 ].SuchmeshoptimizationalgorithmscangivefurtherimprovementsontheoutputofbothouralgorithmsandthepreviousDelaunayrenementalgorithms. 84

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85

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DEF Figure3-25. Comparingsmoothingalgorithmswithourmethod.A)Triangle1.4(circumcenter),[,]=[30,118],1073points,83msec.B)SmartLaplaciansmoothingappliedon(A),[,]=[31.2,110.8],348msec.C)Angle-basedsmoothingappliedon(A),[,]=[24.2,109.8],346msec.D)Algorithm 3 ,[,]=[40,99.8],1086points,40msec.E)SmartLaplaciansmoothingappliedon(D),[,]=[40,99.3],347msec.F)Angle-basedsmoothingappliedon(D),[,]=[20.5,102],377msec. 86

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6 105 ].Alowerboundontheanglesofatriangulationimpliesanimmediateupperbound2onitsangles.Mosttriangulation(meshing)algorithmsandsoftwarerelyonthisobservationandprovidealowerboundontheangles.Whileformanyapplications,suchanupperbound(immediatefromalowerbound)servesaspartofasatisfactorysolution,therearemanycasesitleadstosub-optimalnumericalsolutionsorevenworse,donotmeettherequirementsofthenumericalformulationinuse.Therearetriangulationalgorithmsprovidingdirectlyanupperboundontheangles[ 9 12 13 15 49 86 ].However,ingeneralanupperbounddonotimplyalowerboundontheanglesofatriangulation.Exceptonlyafewstudiesaddressingtheproblem[ 7 82 ],therehasbeenalackoftriangulationmethodsandsoftwarethatprovidestronglowerandupperboundsontheanglessimultaneously.Here,weproposeamethodthattriangulatescomplexdomainssuchthatallanglesareinanintervalasnarrowas[40,86],addressingProblem 1 inSection 2.3 Aqualitativecomparisonofourmethodagainstthepreviousmethods,wherewelabeltheperformanceofeachmethodonascaleof4,as1.Poor,2.Fair,3.Good,or4.Premium. ClassicalDelaunayLOSPWCTAcuteProposedrenementmethodmethodmethodmethodFeature[ 99 104 ][ 51 ][ 120 ][ 50 ] TimePerformanceGoodGoodPoorFairGoodHandlingcomplexinputGoodGoodFairGoodGoodNosmallangleGoodPremiumFairFairPremiumNolargeangleFairFairGoodGoodPremiumNolargenosmallangleFairFairFairFairPremium 87

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Acutetriangulation([,]=[25,85])ofthedomainAcute(inPalatinofont)withnosmallangle<25. Table 4-1 givesapracticalperformancecomparisonoftheproposedmethodandfouravailableimplementations,whereeachreliesonadifferentapproach.Notethat,quadtree-basedmethodsareomitted,sincetheyareexpectedtogeneratelargetriangulationsinpractice.QualitativelabelsassignedinthistableandrelatedcontributionsarejustiedbytheexperimentalstudygiveninSection 4.4 .ContributionofChapter 4 canbesummarizedas: 28 45 99 104 118 ]andalsothetime-efcientDelaunayrenementalgorithmsof[ 68 84 ].First,weemployagenericoptimization-basedSteinerpointinsertionprocedureratherthanusingaparticulartypeofSteinerpoint(suchascircumcenters,sinks,oroff-centers).Second,weintegratealocalpointrelocationprocedureintotherenement(inordertoavoidtheterminationproblemwhichisacommonundesirednever-endingrenementsyndromewhenstrongangleconstraints 88

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3 andin[ 53 ].Here,weconsiderboththemaximumangleandtheminimumangleconstraints(simultaneously)anddescribeanewSteinerpointinsertionstrategyandanewpointrelocationprocedure.Next,weintroducegeometricconceptsthatenableusintegratingthemaximumangleconstraintintotherenementandtherelocationstepsofourmodiedDelaunayrenementmethod. 6 13 15 105 ].Existingmethods,however,generallyprovideeitheraloweroranupperboundontheangles(andnotstrongenoughlowerandupperbounds).Here,ourgoalistodesignamethodandasoftwarefortriangulatingcomplexgeometryprovidingstronglowerandupperboundsonanglessimultaneously.This,inturn,leadstoareductionofnumericalandinterpolationerrorsthatappearintheuseoftriangulationsinscienticcomputing.Amongseveralversionsoftheangleboundedtriangulationproblems,twoareworthtoelaborateastheyaskforrelativelystrongupperboundsonangles:thenon-obtusetriangulationproblemandtheacutetriangulationproblem.Therearequiteafewscienticandgraphicsapplicationsstatedintheliteraturemotivatingthesetwoproblems.Examplesoftheseapplicationsincludethespace-timediscontinuousGalerkinmethods[ 119 ]andgeodesicpathcomputationonmanifolds[ 71 96 ],meshembeddingviadiscreteHarmonicmaps[ 42 ],discreteexteriorcalculusmethods[ 120 ]anddiscretemaximumprincipleonlinearniteelementsolutions[ 72 ].Notethat,thelowertheupperboundonlargeangles,thehighertheefciency,accuracyandvalidityoftheapplications.Inparticular,theymostlyrequireacutetriangulations.Ingeneral,havinganadditionallowerboundontheanglesisalsoimplicitlyimportantintheseapplications. 89

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2.3 ,therearemanyalgorithmsforcomputingtriangulationswithnosmallangles,e.g.,thequadtree-basedmethods[ 13 ],theDelaunayrenementmethods[ 53 68 84 99 104 ],andtheedgerenementmethods[ 98 ].AmongthesearguablythebestperformanceisduetoDelaunayrenementmethods,whichinpracticeworkforashighas34.They,however,sufferfromaterminationproblemevenforsimplegeometricinputdomainswheneverthedesiredlowerboundis35orlarger.Onlyrecently,ournewDelaunayrenementalgorithmgiveninChapter 3 andin[ 53 ]thatreliesonlocallyoptimalSteinerpointsisshowntoworkinpracticeforashighas42.Suchanboundcankeepallanglesintheinterval[42,96],however,itisnotstrongenoughforcomputingacuteornon-obtusetriangulations.So,eventhebestexistingrenementmethodsdonotgiveasatisfactorysolutionforProblem 1 .Here,weelaboratemoreonlocallyoptimalSteinerpointconcepttogetherwiththelocalrelocationstrategyandprovidetriangulationswithoutsmallandlargeangles. 4-2 .Considerthecirclethatgoesthroughp,qandathirdpointysuchthatyandrareonthesamesideofpqand\pyq=.Thediskboundedbythiscircleiscalled-petal(pq).Similarly,considerthecirclethatgoesthroughp,qandathirdpointzsuchthatzandrareonthesamesideofpqand\pzq=.Thediskboundedbythiscircleiscalled-petal(pq).Then,thedifferenceofthesets-petal(pq)andtheinsideofthe-petal(pq)isdenedas[,]-crescent(pq)(seeFigure 4-2 (A)).Let-slab(pq)denotetheregionbetweenthetwolinesonegoingthroughp,theotherqandbothmakinganangletolinesegmentpq.Similarly,let-wedge(pq)denotetheregionoutsidethetwolinesonegoingthroughp,theotherqandbothmakinganangletolinesegmentpq.Then,[,]-slice(pq)isdenedastheintersection 90

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4-2 (B)). AB Figure4-2. A)[,]-crescent(pq)isshown.B)Feasibleregion[,]-slice(pq)istheintersectionof[,]-crescent(pq),-wedge(pq),and-slab(pq)apointxinsidethisregionthatisfurthestawayfromallexistingverticesischosenasarenementpoint. Hence,forpointinsertionprocedure,wesuggesttoinsertaSteinerpointinthesliceregionofabadtrianglethatisfurthestawayfromallexistingvertices.Notethat,thispointisnotnecessarilyaVoronoivertexoronaVoronoiedge.However,itiseitheronVoronoiskeletonoronthesliceboundary.ThisisgiveninLemma 3 .Therefore,asimplelocalsearchontheVoronoidiagramhelpustondthatpointinsidetheslice.Alsoobservethat,whilethe-wedgeandthe-petalensuretheminimumangleconstraint,the-slabandthe-petaldothesameforthemaximumangleconstraint.ThefollowinglemmaissimilartoLemma 1 inChapter 3 wherethesearchregionisconstructedconsideringonlythe-petal(pq).Thedistinctionhereisduetotheadditionofthemaximumangleconstraint(imposedby-slab(pq)ontheviewangleand-slab(pq)onthesideangles)andalsotheminimumangleconstraint(imposedby-slab(pq))onthesideangles.Thankstotheadditionoftheseconstraintstheregionwe 91

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53 ],theregionisnotnecessarilyconvex. AB Figure4-3. AdditionaltypesofSteinerpoints.A)Slabintersectioncase.B)Slab-petalintersectioncase. InChapter 3 ,wedenealocallyoptimalSteinerpointtobeonaVoronoiedgeoraVoronoivertexandgivefourdifferenttypesofSteinerpointsaccordingly.Here,dueto 92

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4-3 demonstratesthesenewtypesofSteinerpointsastheintersectionofaVoronoiedgeandaslabline,andtheintersectionofaslablineandthepetal. Figure4-4. Abadtrianglepqaisxedbyrelocatingoneofitsverticesaintobwhichisapointintheintersection(shaded)ofthe[,]-slicesoftheedgesonthelinkofa. Thestarofavertexaconsistsofalltrianglesthatcontaina.Thelinkofa,then,consistsofallsurroundingedgesoftrianglesinthestarthatarenotincidenttoa.A 93

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4-4 Figure4-5. Theintersectionoftheslicesoflinkedgesisnotnecessarilyconvexnorasingleconnectedcomponent.Forclarity,weomitthe-wedgeand-petalconstraints;afterallaweakconstraintwouldresultinthesamefeasibleregion. Figure4-6. Approximateconvexsupersetofasliceregion.Forefcientimplementation,feasibleintersectionregionforrelocationiscomputedbyusingapproximatesliceregions. 94

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(DelTri)oftheinput LetPdenotethemaintainedpointset ifrelocated==FALSEthen fromallexistingvertices 4-5 whichillustratesafeasiblepointrelocationregionwiththreeindependentcomponents.Havingcurvedboundarysegmentsalsoaddsdifcultytothecomplexityofcomputingfeasibleregions.Inordertoeasethesecomplexities,wecomputeaconvexapproximatesupersetofthefeasibleregion.Thecurvedboundarysegmentduetoeach-petalconstraintisapproximatedwith6linearsegments.Inaddition,wepostponeimposingthe-petalconstraints.SeeFigure 4-6 .Oncewecomputeaconvexapproximatesupersetofthefeasibleregion,wedoasamplinginsidethisregionandcheckwhetherthereisasamplepointthatmakesallnewtrianglesincidenttoitgood.Algorithm 4 summarizestheintegrationoftherenementandtherelocationsteps. 95

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4.4.3 .AcomparativestudyagainsttheexistingrenementmethodsandsmoothingmethodsaregiveninSections 4.4.4 and 4.4.5 .WealsogenerateuniformtriangulationswithoutsmallandlargeangleswiththeadditionalmaximumareaconstraintinSection 4.4.6 3 [ 53 ],whichintegratesapointrelocationstepintotheDelaunayrenement.Themodications,however,aresubstantial,aswedealwiththecomplexitiesarisingfromthenewconstraints-wedge,-petaland-slab.Intheinsertionstep,weprimarilyenforcethe-petalandthe-slabconstraints,anddoalazyevaluationofthe-petalandthe-wedgeconstraints.Weperforma(graph)searchontheVoronoiskeletonandtheboundaryofthesliceregiontolocatealocallyoptimalSteinerpoint.Intherelocationstep,weprimarilyenforcethe-petal,the-wedge,andthe-slabconstraints,anddoalazyevaluationofthe-petalconstraint.Replacingthecurvedboundarysegmentsofthe-petalswithpiecewiselinearsegmentsenablesustousethestandardhalf-planeintersectionalgorithmforcomputingaconvexapproximatesupersetoftheintersectionofslices.Then,weperformauniformsamplingonthissupersetregioninordertolocateanactualfeasiblepointforrelocation. 4-2 .Forthegeometricdomains 96

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99 104 ].Throughoutthischapter,triangulationsarevisualizedusingthetrianglecoloringschemedescribedinFigure 4-7 .Thisway,badtriangleswhoseanglesarenotintheinterval[,]standoutintheoutputvisualization. Gray(Light)Scale Red(Dark)Scale Figure4-7. Thegray(light)andthered(dark)scalesareusedforcoloringthegoodandthebadtriangles,respectively.Ineachcase,thelargestangleofthetriangledeterminestheparticularcolor. 4-8 illustratethatouralgorithmiscapableofhandlingalmostallofthese800+[,]combinationsexceptwhenbothandareverystrongconstraints,e.g.[,]=[40,80].Itisevidentinthisplot(andalsoinTable 4-2 andFigure 4-14 )thatasweenforcestrongerminimumangleand/ormaximumangleconstraintsthesizeoftheoutputtriangulationincreases.ThisphenomenonisalsoshowninFigures 4-9 and 4-10 .UpperpartoftheplotsillustratestherelationshipbetweentheminimumangleconstraintandthenumberofSteinerpointsinsertedforaconstantmaximumangleconstraintandviceversa.Concurrently,lowerpartoftheplotsreportsthesamefortherunningtime,whichrevealsaparallelbehavior.Theplotsfortheotherdatasetsaresimilar.Actualoutputtriangulationsandtheirhistogramofangles 97

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4-11 and 4-12 .Here,differentlevelsofqualitytriangulationsareshownfromgoodqualitytopremium. Figure4-8. PlotofoutputsizeforvaryingandforLakeSuperiorandTurkeydatasets. 98

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PlotofoutputsizeandtimeperformanceofthealgorithmbasedonaxedminimumangleconstraintandavariablemaximumangleconstraintforTurkey(=20)andKoala(=30)datasets. 99

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PlotofoutputsizeandtimeperformanceofthealgorithmbasedonaxedmaximumangleconstraintandavariableminimumangleconstraintforCalifornia(=85)andBoeing(=85)datasets. 100

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B C D Figure4-11. OutputtriangulationsandtheiranglehistogramsofChinadatasetforvariousminimumandmaximumangleconstraints.A)[,]=[1,90],849points.B)[,]=[10,89],889points.C)[,]=[20,87],1073points.D)[,]=[30,85],1409points. 101

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B C D Figure4-12. Outputtriangulationsandtheiranglehistogramsofgoldshdatasetforvariousminimumandmaximumangleconstraints.A)[,]=[5,85],1497points.B)[,]=[15,85],1454points.C)[,]=[25,85],1584points.D)[,]=[35,85],2249points. 102

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Performanceofthealgorithmforfourdifferent[,]rangeson15differentdatasets.(Continuedonthenextpage.) InputOutputTimeDataset#Points#Segments#Holes[,]#Points(msec) Cyprus4484470[30,90]1540211[35,83]89181810[40,86]97551270[25,81]166786001Turkey2162160[30,90]57162[35,83]1796341[40,86]4496608[25,81]89733412Italy2162160[30,90]489142[35,83]1293225[40,86]47041256[25,81]32691395Iraq1921920[30,90]466249[35,83]2345472[40,86]3475621[25,81]72172903Florida3043040[30,90]895139[35,83]3645705[40,86]4084516[25,81]89013233Superior5225227[30,90]1250157[35,83]3965751[40,86]5308661[25,81]75322701Fish92921[30,90]205258[35,83]52899[40,86]2399356[25,81]28451043Dolphin2002001[30,90]45664[35,83]1339283[40,86]1430166[25,81]31651130Alligator110011003[30,90]4205618[35,83]156023086[40,86]214252669[25,81]4767817960Hexagon860[30,90]275[35,83]467[40,86]536[25,81]279 103

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4-2 :Continued. InputOutputTimeDataset#Points#Segments#Holes[,]#Points(msec) Boeing30301[30,90]15851[35,83]640152[40,86]1053145[25,81]22791035BoxedPair640[30,90]5734[35,83]431108[40,86]21931[25,81]1364730Kuzmin200000[30,90]62691322[35,83]190864247[40,86]290864185[25,81]6166123922Plate65645[30,90]24945[35,83]2718616[40,86]3664501[25,81]49101865Random100000[30,90]3426761[35,83]120622807[40,86]154102345[25,81]3112212108 Table 4-2 presentstheperformance(timeandoutputsize)ofouralgorithmon15datasetsforfourdifferentcombinationsof[,]values.Itshouldbenotedthatouralgorithmhandlesthe[,]=[30,90]constraintwithgreatease,resultinginafastcomputationofsmallsizehighqualitynon-obtusetriangulations.Timeperformanceisstillreasonableforsomewhatstrongerconstraints,e.g.[,]=[35,83].However,pushingeitheroneoftheseconstraintsfurtherstrongerrequiresaslightrelaxationoftheotherconstraintinordertoassuretheterminationofthealgorithm.Forinstance,theinterval[25,81](verystrongmaximumangleconstraint)andtheinterval[40,86](verystrongminimumangleconstraint)aretwo[,]constraintsforwhichouralgorithmisshowntoworkonalltheexperimenteddatasets.Theresultspresentedhereshouldbetakenasatestamenttothebroadcapabilitiesofourmethodandthesoftware. 104

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ComparisonofourmethodwiththepreviousDelaunayrenementmethods(classicalDelaunayrenementmethod[ 28 45 99 104 118 ]andLOSPmethod[ 53 ])intermsoftheoutputsize(numberofpoints)andtheanglerange[,]. DataClassicalDel.Ref.LOSPMethodOurMethodset[,]#Pts[,]#Pts[,]#Pts 105

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4-3 comparesclassicalDelaunayrenementmethod[ 28 45 99 104 118 ],LOSPmethod[ 53 ]andourmethodintermsofmaximumandminimumangleaswellasoutputsizesofthetriangulations.OurexperimentsshowthatclassicalDelaunayrenementmethodgivesmaximumanglesaslowas110,whereasLOSPmethodreducethisfurtherto98.Although,thesealgorithmscanprovidelargesmallestangles,thisdoesnotyieldtosmalllargestangles.Ontheotherhand,ourmethod,fornearlythesamenumberofoutputpoints,canachieveminimumanglesashighas35whilekeepingthemaximumanglesaslowas85.Forlargervalues,withoutsacricingfromtheoutputsize,wecanreachsmallervaluesascomparedtopreviousDelaunayrenement-basedmethods.Signicantdifferenceinrangevaluesindicatesthestrengthofourmethod,suchthatclassicalDelaunayrenementmethodandLOSPmethodprovidesintervalrangesaround75and57,respectively,whereasthisvalueisaround49forourmethod.Figures 4-13 and 4-14 illustratetheoutputoftheclassicalDelaunayrenementandourmethod,respectively,forvarying[,]constraints.NotethatclassicalDelaunayrenementtakesonlytheconstraintasinput.InFigure 4-13 ,thereareindeedonlythreedifferenttriangulationsoneforeachof=11,22,33.Forcomparison,wecoloredthebadtriangleswithrespecttothecorrespondingconstraintsimposedintheimplementationoftheproposedmethod.Outofthetwelvedifferentcombinationsof[,]constraints,theclassicalDelaunayrenementmethodgivesanoutputacceptablebyonlyonecombination([33,115]),wherethisismainlybecausetheupperboundforisactuallyinthetriviallargeanglebound,180233=114.Ontheotherhand,ourmethodsuccessfullygeneratesqualitytriangulations(Figure 4-14 )forallcorresponding[,]values.(SeealsoFigure 4-17 .) 106

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OutputoftheclassicalDelaunayrenementmethodfor=11,22,and33.Trianglesaremarkedaccordingtofourdifferentconstraints. 107

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Outputofourmethodbasedonmaximumandminimumangleconstraints[,]. 108

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SizecomparisonofalldatasetsinTable 4-2 for[,]=[30,120]intervalbyapplyingclassicalDelaunayrenementmethod,LOSPmethodandourmethod. InFigure 4-15 ,weplottheratioofthenumberofSteinerpointsinsertedbytheDelaunayrenementmethodandourmethodaveragedover15datasets.Forthiswenaturallyenforcedan[,]intervalwhichcanbeachievedbytheclassicalDelaunayrenementandLOSPmethods.TheoutputsizeofourmethodisroughlythesameastheLOSPmethod,whereaswegeneratesignicantlysmallertriangulationsthanclassicalDelaunayrenementmethod.ThisisinagreementwiththeobservationmadeinTable 4-3 thatourmethodgeneratesbetterqualitytriangulationsusingnearlythesamenumberofpoints.Also,Figure 4-16 givesanoutputcomparisonexampleforareasonablyhighandatrivial=2constraint. 120 ]whichisdesignedtoiterativelysmoothaninputtriangulationintoanacutetriangulation.Forthisexperiment,weprovidetheoutputofaclassicalDelaunayrenementmethod(Figure 4-17 (A)),asaninputtotheWCTmethod.AsseeninFigure 4-17 (C),WCTmethodimproves 109

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120 ].WealsoincludeanoutputoftheLOSPmethodforcomparisoninFigure 4-17 (B).Usingnearlythesamenumberoftrianglesasthepreviousmethods,ourmethodcomputestriangulationswithmuchbetterqualitykeepingallanglesin[35,85]asignicantimprovementovertheWCT's[25,97].Moreover,theproposedmethodismorethananorderofmagnitudefasterthantheWCTmethodandhascomparablespeedtotheLOSPandtheclassicalDelaunayrenementmethods.Comparativeexperimentsonotherdatasetsrevealsimilarresults. AB Figure4-16. ComparingoutputsizesandanglehistogramsofpreviousclassicalDelaunayrenementalgorithmandoursforthesame[,]=[30,120]interval.A)ClassicalDel.Ref.[ 99 118 ],759points.B)ClassicalDel.Ref.[ 99 118 ],588points. 110

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CD Figure4-17. Iraqmap.Outputof(A)theclassicalDelaunayrenementfor=33,(B)theLOSPmethodfor=41,(C)theWCTmethodwhere(A)isusedasinput,and(D)theproposedmethodfor[,]=[35,85].Triangleswithanangleoutsidetheinterval[,]=[35,85]aremarkedasbad.A)ClassicalDel.Ref.[ 99 118 ](Initial),[,]=[33.021,112.7],1155points,66574msec.B)LOSPMethod[ 53 ],[,]=[41,97.7],1007points,185msec.C)WCTMethod[ 120 ],[,]=[24.918,97.711],1155points,65msec.D)OurMethod[,]=[35.15,85],971points,38msec. 111

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Figure4-18. PremiumqualityoutputtriangulationsforBoeingandhexagoninputdomainswiththeadditionalmaximumareaconstraint.A)[,]=[38,88],maxarea=0.003,shortestedge=0.0036.B)[,]=[40,86],maxarea=0.05,shortestedge=4. 4-18 and 4-19 ,wegiveoutputtriangulationsfordifferentdatasetsforvarious[,]intervalsandmaximumareaconstraints.WhileFigure 4-18 showsthestrengthofourmethodtoproduceuniformtriangulationswithoutsmallandlargeanglesforsignicantlygoodintervals,Figure 4-19 comparesuniformtriangulationsfordifferentsizeconstraintsforthesameangleranges. 112

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Figure4-19. PremiumqualityoutputtriangulationsforTurkeydatasetwithdifferentmaximumareaconstraints.A)[,]=[35,85],maxarea=100,shortestedge=3.6.B)[,]=[35,85],maxarea=200,shortestedge=3.6. 113

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52 ].Websitealsoincludesthedescriptionsofourideasandgivereferencestoourpublishedwork.Moreover,adetailedcomparisonwithTriangle,whichisoneofthemostpopulartriangulationsoftware,isgiven.Weprovidesampleanimationsofouralgorithmstodescribetheprocessandpointoutimportantdetails/challenges. 114

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52 ].Here,wegiveasummaryoftheimportantdifferencesandsimilaritiesbetweenthem.ThepopularmeshgenerationsoftwareTriangle[ 106 ]isarobustimplementationoftheDelaunayrenementmethod[ 101 ].ThesoftwareusedRuppert'scircumcenterinsertionalgorithm[ 99 ]intherstfourreleases(latestTriangle1.4)andhasbeenusingUngor'soff-centerinsertionalgorithm[ 118 ]initslatesttworeleases(Triangleversions1.5and1.6).Here,wecompareoursoftwarewithTriangleversion1.4,mainlybecauseitisaclassicalrepresentationofDelaunayrenementalgorithmsandalsooff-centersarepartofouruniedSteinerpointdenition.However,comparingwithversion1.6wouldgivesimilarresults,exceptitsadvancingfronttypeofalgorithmicapproach,andhavingslightlybettersmallanglebounds(35)andsmalleroutputsize(nearlyhalf)thancircumcenter-basedalgorithm. 3 and 4 startingfromtheboundary.Conversely,Trianglefollowscircumcenterinsertion,wherenospeciclocationisxedinorder.(SeeFigure 5-1 .) 115

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Improvingtheoreticalbounds.ThetheoreticalterminationandsizecomplexityboundsgivenforthepreviousDelaunayrenementalgorithmsalsoapplyforouralgorithms(constraintanglesupto30),aswearemorecautiousinintroducingshortfeatures.Itwouldbeinterestingtoprovethesametheoretical(terminationandsize-optimality)boundsfor>30.Inaddition,acarefulanalysisonourheuristicmethodgeneratingnosmallnolargeangletriangulationsmightexplainthegoodpracticalperformanceofourmethod.Aguaranteed[,]boundwouldbeusefulforfuturestudies. 104 ].Analternativeboundaryhandlingrulemightbemoreeffective. 34 ].Weforeseethatouralgorithmscanbeeasilyextendedforcomputinghighqualityandsmallsizetriangulationsoftwomanifolds.Extensionofthemethodtothreedimensionsisalsoanaturalresearchdirection. 55 ].Preliminaryresultsshowedgreatpotentialforapowerfulsmoothingtechnique,especiallyforhandlingcomplexdomainsunlikeothersmoothingmethods.Furtherdevelopingourideasformeshimprovementtechniquescouldresultinsuccessfulapplications. 116

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IJ CD KL EF MN GH OP Figure5-1. AlgorithmicowofTriangleandaCuteforbuffalodatasetfor=34.A-N)Triangle,0,30,50,100,150,200,300,412Steinerpointsinserted,respectively.I-P)aCute,0,30,50,75,85,105,110,114Steinerpointsinserted,respectively. 117

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] 118

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[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] 119

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[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] 120

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[42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] 121

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[55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] 122

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[69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] 123

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[83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] 124

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[95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] 125

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[108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] 126

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[121] [122] [123] [124] [125] 127

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HaleErtenreceivedherBachelorofSciencedegreeinComputerEngineeringfromtheBogaziciUniversity,IstanbulTurkeyin2005.ShepursuedherMasterofScienceandDoctorofPhilosophydegreesattheUniversityofFloridafromthedepartmentofComputerandInformationScienceandEngineeringuntilhergraduationin2009.Herresearchinterestsincludemeshgeneration,computationalgeometry,algorithmsandgraphics. 128