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  • AB HELSINKI UNIVERSITY OF TECHNOLOGYFACULTY OF INFORMATION AND NATURAL SCIENCESDEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

    Using Delaunay Triangulation in

    Infrastructure Design Software

    Masters Thesis

    Ville Herva

    Department of Computer Science and Engineering

    Espoo 2009

  • ii

    AB TEKNILLINEN KORKEAKOULUINFORMAATIO- JA LUONNONTIETEIDEN TIEDEKUNTATIETOTEKNIIKAN LAITOS

    Diplomityn tiivistelm

    Tekij: Ville Herva

    Tyn nimi: Delaunay-kolmioinnin hydyntminen infrastruktuurin

    suunnitteluohjelmistoissa

    Pivys: 28.01.2009 Sivuja: 80+11

    Tiedekunta: Informaatio- ja luonnontieteiden tiedekunta

    Professuuri: T-106

    Tyn valvoja: prof. Jorma Tarhio

    Tyn ohjaaja: DI Timo Ruoho

    Infrastruktuurin suunnitteluohjelmistoissa, kuten tien-, rautatien-, sillan-, tunnelin-, ja

    ympristnsuunnitteluohjelmistoissa, on Suomessa perinteisesti kytetty maaston pinnan

    mallintamiseen mittapisteist muodostettua epsnnllist kolmioverkkoa. Muualla maailmassa

    ovat kytss olleet snnlliset neli- ja kolmioverkot, maaston approksimointi ilman

    pintaesityst, sek joissain tapauksissa algebralliset pintaesitykset.

    Pinnan approksimaatiota tarvitaan em. sovelluksissa mm. pisteen korkeuden arviointiin, 2-

    ulotteisten murtoviivojen interpolointiin maaston pinnalle, korkeuskyrien laskemiseen ja

    massan (tilavuuden) laskentaan annetuilta alueilta sek visualisointiin.

    Delaunay-kolmiointi on tapa muodosta 2-ulotteisesta pistejoukosta epsnnllinen

    kolmioverkko, jonka kolmiot hyvin tasamuotoisia. Kolmioiden tasamuotoisuus on oleellisesta

    pintamallin tarkkuudelle.

    Tss tyss tutkitaan Delaunay-kolmioinnin kytettvyytt maaston mallintamiseen suurilla

    pistejoukoilla, sek epsnnllisen kolmioinnin kytettvyytt em. tehtviin. Tyss vertaillaan

    Delaunay-kolmioinnin muodostamisen ajan ja muistin kulutusta pintaesityksen

    muodostamiseen muilla menetelmill. Lisksi tutkitaan nin muodostettujen pintamallien

    tilavuuslaskennan ja interpolaation nopeutta ja tarkkuutta.

    Avainsanat: laskennallinen geometria, paikkatieto

    Kieli: englanti

  • iii

    AB HELSINKI UNIVERSITY OF TECHNOLOGYFACULTY OF INFORMATION AND NATURAL SCIENCESDEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

    Abstract of Masters Thesis

    Date: 28.01.2009 Pages: 80+11

    Faculty: Faculty of Information and Natural Sciences and Engineering

    Professorship: T-106

    Supervisor: Professor Jorma Tarhio

    Instructor: Master of Science Timo Ruoho

    In Finland, irregular triangulation has traditionally been used in infrastructural design software,

    such as road, railroad, bridge, tunnel and environmental design software, to model ground

    surfaces. Elsewhere, methods like regular square and triangle network, approximating surface

    without a surface presentation, and algebraic surfaces, have been used for the same task.

    Approximating the ground surface is necessary for tasks such as determining the height of a

    point on the ground, interpolating 2D polylines onto the ground, calculating height lines,

    calculating volumes and visualization.

    In most of these cases, a continuous surface representation, a digital terrain model is needed.

    Delaunay triangulation is a way of forming an irregular triangulation out of a 2D point set, in

    such a way that the triangles are well-formed. Well-formed triangles are essential for the

    accuracy of the surface representation.

    This Masters Thesis studies how much time and memory it takes to form a Delaunay

    triangulation for large point sets, and how Delaunay triangulation compares to other methods

    of forming a surface representation. In addition, the run-time and accuracy of the resulting

    surface representations is studied in different interpolation and volume calculation tasks.

    Keywords: computational geometry, geographical information

    Language: English

  • iv

    Table of Contents

    Table of Contents ......................................................................................................................................................... iv Terminology and Abbreviations ................................................................................................................................ vi List of Figures ............................................................................................................................................................... ix List of Diagrams ............................................................................................................................................................ x List of Tables .................................................................................................................................................................. x 0 ............................................................................................................................................................. xi Foreword1 ......................................................................................................................................................... 1 Introduction2 .............................................................................................................................................. 4 Problem Statement2.1 ............................................................................................................................................... 4 Scope of Thesis2.2 ........................................................................................................ 5 Triangulation of an Irregular Point Set2.2.1 ............................................................................................................................ 7 Arbitrary Triangulation2.2.2 .................................................................................................................... 8 Quality of the Triangulation2.2.3 ............................................................................................................................ 9 Delaunay Triangulation2.2.4 .......................................................................................................................................... 11 Other Criteria2.3 ................................................................................................................................ 13 Alternative Approaches2.3.1 ........................................................................................... 13 Regular Grid (Rectangular) Triangulation2.3.2 ........................................................................ 14 Bzier surfaces and Non-uniform B-Spline surfaces2.3.3 ..................................................................................................................... 15 Direct Point Interpolation3 ............................................................................................ 18 Infrastructure Design Software and Workflow3.1 ....................................................... 20 Input Data Considerations for the Infrastructure Design Process3.2 ............................................................................................................................ 21 Quality of the Input Data4 ................................................................................................................................. 22 Triangulation Algorithms4.1 .................................................................................................................................. 22 Algorithms Categories4.1.1 .............................................................................................................................. 22 Sweepline Algorithm4.1.2 ............................................................................................................ 23 Divide-and-conquer Algorithm4.1.3 ........................................................................................................................ 25 Radial Sweep Algorithm4.1.4 .......................................................................................................................... 27 Step-by-step Algorithm4.1.5 ........................................................................................................................... 28 Incremental Algorithm4.1.6 ............................................................................................................ 31 Convex Hull Based Algorithms4.1.7 ..................................................................................................................................... 32 Specialized Cases4.2 .................................................................................................. 32 Point Data Triangulation with Fold-lines4.2.1 .................................................................................................................................. 34 Other Approaches4.2.2 ........................................................................................................................... 34 Additional Constraints

  • v

    5 ..................................................................................... 35 Manipulating and Using an Existing Triangulation5.1 ............................................................................................................................................... 35 Adding a Point5.2 .................................................................................................................................. 36 Adding a Set of Points5.3 ......................................................................................................