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Numerical Stabilization of Geometric Algorithms and Its Applications

Research Project

Project/Area Number 01550279
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 計算機工学
Research InstitutionUniversity of Tokyo

Principal Investigator

IMAI Toshiyuki (1990-1991)  University of Tokyo, Department of Mathematical Engineering and Information Physics, Assistant, 工学部, 助手 (90213214)

杉原 厚吉 (1989)  東京大学, 工学部, 助教授 (40144117)

Co-Investigator(Kenkyū-buntansha) TOMIOKA Yutaka  University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 助手 (30188776)
SUGIHARA Kokichi  University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 教授 (40144117)
IRI Masao  University of Tokyo, Department of Mathematical Engineering and Information Phys, 工学部, 教授 (40010722)
今井 敏行  東京大学, 工学部, 助手 (90213214)
Project Period (FY) 1989 – 1991
Project Status Completed (Fiscal Year 1991)
Budget Amount *help
¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 1991: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1990: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1989: ¥900,000 (Direct Cost: ¥900,000)
KeywordsGeometric algorithm / Topological consistency / Voronoi diagram / Numerical error / ombinatorial geometry / 数値誤差対策 / 位相優先法 / 組合せ幾何学 / 数値的安定化 / 逐次添加法 / 数値的矛盾 / 幾何的退化
Research Abstract

Conventionally, geometric algorithms have been designed on the assumption that numerical errors do not take place, and consequently they are not necessarily valid in real computation where numerical errors are inevitable. We searched for a method for designing geometric algorithms that are robust against numerical errors, and established a new design methodology in which the highest priority is placed on the topological consistency of geometric objects and numerical results are used as lower-priority information. Algorithms thus designed does not come across inconsistency and hence carries out its task, giving some output that is at least topologically consistent. This design methodology was applied to design robust algorithms for the problems : (1) the incremental construction of two-dimensional Voronoi diagrams for points, (2) the divide-and-conquer construction of two-dimensional Voronoi diagrams for points, (3) the incremental construction of two-dimensional Voronoi diagrams for li … More ne segments, (4) the incremental construction of three-dimensional Voronoi diagrams for points, (5) the extraction of intersections of line segments, (6) the construction of three-dimensional convex hulls of points, (7) the set-theoretic operations of simple polygons, (8) the intersection of convex polyhedra, and (9) the construction of Laguerre Voronoi diagrams in the plane. In particular, the new algorithms for the problems (1)-(5) were implemented into computer programs, and computational experiments were corried out. The experiments show that, no matter how poor the precision in computation may be, the programs do not fail, that the outputs converge to the true solutions as the precision becomes higher, that the new algorithms achieve robustness without increasing time complexity achieved by conventional algorithms, and that the new algorithms are simple because exceptional processing for degenerate cases is not necessary.
We also investigated numerical methods required in geometric algorithms. The fast automatic differentiation method were used to obtain the tight bound of numerical errors, and the method was applied to many problems such as geographic optimization based on Voronoi diagrams. A new method for tracing abgebraic curves was also constructed, which can trace both global and micro structures in reasonable time. We could also improve the performance of the programs for the problems (1)-(5) bytuning the ways of numerical computations. Less

Report

(4 results)
  • 1991 Annual Research Report   Final Research Report Summary
  • 1990 Annual Research Report
  • 1989 Annual Research Report
  • Research Products

    (72 results)

All Other

All Publications (72 results)

  • [Publications] K.Sugihara: "Two design principles of geometric algorithms in finite-precision arithmetic" Applied Mathematics Letters. 2. 203-206 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原厚吉: "計算幾何学的手法と画像解析-ボロノイ図の応用を中心として" 情報処理. 30. 1067-1075 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "On finite-precision representations of geometric objects" Journal of Computers and System Sciences. 39. 236-247 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A solid modelling system free from topological inconsistercy" Journal of Infomation Processing. 12. 380-393 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原厚吉: "図形処理における数値的諸問題とその対策" 東京大学工学部紀要. A-27. 40-41 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Kishimoto: "A practical approach to the definition and measurement of “length"" Japan Journal of Applied Mathematics. 6. 179-207 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 伊理正夫: "数値計算の基礎技術の最近の話題から-ノルム、丸め誤差、偏微分と高速自動微分法を中心として" 電子情報通信学会誌. A 72. 1044-1052 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 久保田光一: "区間演算を用いた丸め誤差解析" 情報処理. 31. 1212-1219 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 大石泰章: "数値的に安定な分割統治型Voronoi図構成算法" 情報処理学会論文誌. 32. 709-720 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原厚吉: "幾何的アルゴリズムの数値的破綻とその対策" 応用数理. 1. 280-299 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Norms,rounding errors,partial derivatives and fast automatic differentiation" IEICE Transactions. E74. 463-471 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 伊理正夫: "高速自動微分法(I);(II)" 応用数理. 1. 17-35 153-163 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Geometrical/geographical optimization and fast automatic differentiation" Yugoslav Journal of Operations Research. 1. 121-134 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "History of automatic differentiation and rounding error estimation" SIAM Automatic differentiation of algorithms-Theory,Implementations,and Application. 3-16 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction" IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences. (掲載予定). (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A robust intersection algorithm based on Delaunay triangulation" IEEE Computer Graphics and Applications. 12. 59-67 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Voronoi diagrams in a river" International Journal of Computational Geometry and Applications. (掲載予定). (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原厚吉: "投影図の自動理解へ向けて" 数理解析研究所講究録. 699. 116-131 (989:)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 今井敏行: "組合せ構造を優先した線分ボロノイ図の構成法" 情報処理学会研究報告(アルゴリズム研究会). 89-AL-11-2. 1-8 (1989)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 稲垣宏: "3次元ボロノイ図構成算法の数値的安定化" 情報処理学会研究報告(アルゴリズム研究会). 91-AL-24-6. 1-8 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Topology-oriented approach to robustness and its applications to several Voronoi-diagram algorithms" Proceedings of the 2nd Canadian Conference in Computational Geometry. 36-39 (1990)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] T.K.Dey: "On good triangulations in three dimensions" Proceedings of the ACM Symposium on solid Modeling Foundations and CAD/CAM Applications. 431-441 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Applications of the Delaunay triangulations to geometric intersection problems" Abstracts of the 15th IFIP Conference on System Modeling and Optimization. 82-83 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Construction of the Voronoi diagram for “one million" generators in single-precision arithmetic" Proceedings of the IEEE. (to appear). (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原厚吉: "伊理(監)、腰塚(偏)、他「計算幾何学を地理情報処理」(改訂版)の中の一つの章「幾何的アルゴリズムの数値誤差対策」" 共立出版(発行予定), (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] A.Okabe: "Spatial Tessellations-Concepts and Applications of Voronoi Diagrams" John Wiley and Sons,Inc.(発行予定), (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Two design principles of geometric algoritms in finite-precision arithmetic" Applied Mathematics Letters. vol.2. 203-206 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "On finite-precision representations of geometric objects" Journal of Computers and System Sciences. vol.39. 236-247 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A solid modelling system free from topological inconsistency" Jouranal of Information Processing. vol.12. 380-393 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Kishimoto: "A practical approach to the definition and measurement of "length"" Japan Journal of Applied Mathematics. vol.6. 179-207 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Norms, rounding errors, partial derivatives and fast automatic differentiation" IEICE Transactions. vol.E74. 463-471 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Geometrical/geographical optimization and fast automatic differentiation" Yugoslav Journal of Operations Research. vol.1. 121-134 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "History of automatic differentiation and rounding error estimation" SIAM Automatic differentiation of algorithms-Theory, Implementations and Application. 3-16 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction" IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences. (to appear). (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "A robust intersection algorithm based on Delaunay triangulation" IEEE Computer Graphics and Applications. vol.12. 59-67 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Voronoi diagrams in a river" International Journal of Computational Geometry and Applications. (to appear). (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Topology-oriented approach to robustness and its applications to several Voronoi-diagram algorithms" Proceedings of the 2nd Canadian Conference in Computational Geometry. 36-39 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] T.K.Dey: "On good triangulations in three dimensions" Proceedings of the ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications. 431-441 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Applications of the Delaunay triangulations to geometric intersection problems" Abstracts of the 15th IFIP Conference on System Modeling and Optimization. 82-83 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Construction of the Voronoi diagram for "one million" generators in single-precision arithmetic" Proceedings of the IEEE. (to appear). (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] A.Okabe: "Spatial Tessellations-Concepts and Applications of Voronoi Diagrams (to appear)" John Wiley and Sons, Inc.(1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Computational-geometry methods and image analysis-Concentrated on applications of Voronoi diagrams (in Japanese)" Transactions of Information Processing Society of Japan. vol.30. 1067-1075 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Several approaches to numerical problems in geometric algorithms (in Japanese)" Journal of the Faculty of Engineering of the University of Tokyo. vol.A-27. 40-41 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Some recent topics in numerical computation-norm, runding error, partial differentiation and fast automatic differentiation (in Japanese)" Journal of the Institute of Electronics, Information and Communication Engineers. vol.72. 1044-1052 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Kubota: "Rounding error analysis with interval operations (in Japanese)" Official Journal of Information Processing Society of Japan. vol.31. 1212-1219 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] Y.Ooishi: "Numerically robust divide-and-conquer algorithms for constructing Voronoi diagrams (in Japanese)" Transactions of Information Processing Society of Japan. vol.32. 709-720 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: "Approaches to the problem of numerical failure of geometric algorithms (in Japanese)" Bulletin of the Japan Society for Industrial and Applied Mathematic. vol.1. 280-299 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] M.Iri: "Fast automatic differentiation I ; II (in Japanese)" Bulletin of the Japan Society for Industrial and Applied Mathematics. vol.1. 17-35, 153-163 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] T.Imai: "A Combinatorial-structure oriented algorithm for Voronoi diagrams of line segments (in Japanese)" Technical Note of the Special Interest Group on Algorithms of Information Processing Society of Japan. vol.89-AL-11-2. 1-8 (1989)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] H.Inagaki: "Numerically robust algorithm for constructing 3-dimensional Voronoi diagram (in Japanese)" Technical Note of the Special Interest Group on Algorithms of Information Processing Society of Japan. vol.91-AL-24-6. 1-8 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] K.Sugihara: Approaches to numerically robust algorithms for geometric problems (in Japanese) In M.Iri and T.Koshizuka (eds.) ; Computational Geometry and Geographic Information Processing, 2nd edition. Kyoritsu-Shuppan (to appear),

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1991 Final Research Report Summary
  • [Publications] 杉原 厚吉: "幾何的アルゴリズムの数値的確線とその対策" 応用数理. 1. 280-299 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 伊理 正夫: "高速自動微分法(I),(II)" 応用数理. 1. 17-35 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Masao Iri: "Norms,rounding errors,partial derivatives and fast automatic differentiation" IEICE Transactions. E74. 463-471 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Masao Iri: "Geometrical/geographical optimization and fast automatic differentiation" Yugoslav Journal of Operations Research. 1. 121-134 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] A.Griewank and G.F.Corliss(eds.),Masao Iri: "History of automatic differentiation and rounding error estimation" Automatic Differentiation of algorithms‐‐‐Theory,Implementation,and Application,SIAM. 3-16 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Kokichi Sugihara: "Applications of the Delaunay triangulations to geometric intersection problems" Abstracts of the 15th IFIP Conference on System Modelling and Optimization,Zurich. 82-83 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Kokichi Sugihara: "A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction" IEICE Transactions on Fundamentals of Electronics,Communications and Computer Sciences. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] Kokichi Sugihara: "A robust intersection algorithm based on Delaunay triangulation" IEEE Computer Graphics and Applications. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] Kokichi Sugihara: "Voronoi diagram in a river" International Journal of Computational Geometry and Applications. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] 大石 泰章: "数値的に安定な分割統治型Voronoi図構成算法" 情報処理学会論文誌. 32. 709-720 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] Barry Boots: "Spatial Tessellation‐‐‐Concepts and Applications of Voronoi Diagrams" John Wiley and Sons, (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] Masao IRI: "An approach to numericalーcombinatorial algorithms with an example of Voronoi diagram construction." Tagungsbericht “Mathematische Optimierung".Mathematisches Forschungsinstitut Oberwolfach,January 8ー12,1990,Oberwolfach,Federal Republic of Germany.(1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Masao IRI: "Integrability of vector and multivector fields associated with interior point methods." Proceedings of International Symposium on Interior Point Methods for Linear Programming:Theory and practice,January 18ー19,1990,Europa Hotel,Scheveningen,The Netherlands.(1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] Kokichi SUGIHARA,Yasuaki OOISHI and Toshiyuki IMAI: "TopologyーOriented Approach to Robustness and Its Applications to Several VoronoiーDiagram Algorithms." Proceedings of the Second Canadian Conference in Computational Geometry.36-39 (1990)

    • Related Report
      1990 Annual Research Report
  • [Publications] 杉原厚吉: "図形処理における数値的諸問題とその対策" 東京大学工学部紀要. A-27. 40-41 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kokichi SUGIHARA and Masao IRI: "Two design principles of geometric algorithms in finite-precision arithmetic" Applied Mathematics Letters. 2. 203-206 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] 杉原厚吉: "計算幾何学的手法と画像解析-ボロノイ図の応用を中心として" 情報処理. 30. 1067-1075 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kokichi SUGIHARA: "On finite-precision representations of geometric objects" Journal of Computer and System Sciences. 39. 236-247 (1989)

    • Related Report
      1989 Annual Research Report
  • [Publications] Kokichi SUGIHARA and Masao IRI: "A solid modelling system free from topological inconsistency" Journal of Information Processing.

    • Related Report
      1989 Annual Research Report
  • [Publications] Kokichi SUGIHARA and Masao IRI: "Construction of the Voronoi diagram for one million generators in single-precision arithmetic" Proceedings of IEEE.

    • Related Report
      1989 Annual Research Report
  • [Publications] 杉原厚吉: "伊理(監),腰塚(編),他「計算幾何学と地理情報処理」(改訂版)の中の一つの章「幾何的アルゴリズムの数値誤差対策」" 共立出版,

    • Related Report
      1989 Annual Research Report

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Published: 1989-04-01   Modified: 2016-04-21  

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