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Geometric Algebra and its application in discrete geometry and topology

Research Project

Project/Area Number 19F19721
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Review Section Basic Section 60020:Mathematical informatics-related
Research InstitutionNational Institute of Informatics

Principal Investigator

杉本 晃宏  国立情報学研究所, コンテンツ科学研究系, 教授 (30314256)

Co-Investigator(Kenkyū-buntansha) BREUILS STEPHANE  国立情報学研究所, コンテンツ科学研究系, 外国人特別研究員
Project Period (FY) 2019-07-24 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2021: ¥200,000 (Direct Cost: ¥200,000)
Fiscal Year 2020: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2019: ¥1,000,000 (Direct Cost: ¥1,000,000)
Keywords幾何操作 / geometric algebra / コンピュータグラフィックス / デジタル幾何 / ディジタル幾何
Outline of Research at the Start

Geometric Algebraは点、直線、曲線、曲面などの対象を統一した形で記述する抽象度の高い数学モデルであるが、これまで、その実利用は困難であるとされてきた。本研究課題は、計算機内の点や直線、曲線、曲面は離散データで管理されているという実践的な視点に立って、そういったデータに対する幾何操作可能なモデルを考案し、実践的に利用可能なGeometric Algebraを構築することを目指す。これによって、Geometric Algebraを用いた幾何計算を計算機内で実現することが可能となり、コンピュータグラフィックスやコンピュータビジョンといった分野に強力な数学的道具を提供することになる。

Outline of Annual Research Achievements

Geometric Algebraは点、直線、曲線、曲面などの対象を統一した形で記述する抽象度の高い数学モデルであるが、これまで、その実利用は困難であるとされてきた。本研究課題では、計算機内の点や直線、曲線、曲面は離散データで管理されているという実践的な視点に立って、そういったデータに対する幾何操作可能なモデルを考案し、実践的に利用可能なGeometric Algebraを構築することを目指している。

昨年度に引き続き、離散データ上に定義された離散曲線や離散局面に対する幾何操作の性質を明らかにするために、格子点上の点集合として表現された対象物に対して、全単射を保証する鏡像変換や回転、さらには剛体変換の性質についての検討を推し進めた。そして、全単射を保証しない鏡像変換、回転、剛体変換を全単射を保証する鏡像変換、回転、剛体変換で近似するアルゴリズムを考案した。また、全単射を保証することと、離散的な意味での位相変化との関係に関しても検討を加えた。さらに、本研究課題でのこれまでの研究成果を発展させる方向性についても検討した。

Research Progress Status

令和3年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和3年度が最終年度であるため、記入しない。

Report

(3 results)
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • Research Products

    (20 results)

All 2021 2020 2019 Other

All Int'l Joint Research (3 results) Journal Article (6 results) (of which Int'l Joint Research: 5 results,  Peer Reviewed: 5 results,  Open Access: 1 results) Presentation (11 results) (of which Int'l Joint Research: 4 results,  Invited: 1 results)

  • [Int'l Joint Research] University of Paris-Est/University of Poitiers(フランス)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] University of Paris-Est/University of Poitiers(フランス)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] University of Poitiers/University of Paris-Est(フランス)

    • Related Report
      2019 Annual Research Report
  • [Journal Article] Visiting Bijective Digitized Reflections and Rotations using Geometric Algebra2021

    • Author(s)
      Stephane Breuils, Akihiro Sugimoto, Yukiko Kenmochi
    • Journal Title

      Proc. of IAPR International Conference on Discrete Geometry and Mathematical Morphology (DGMM2021)

      Volume: LNCS 12708 Pages: 242-254

    • DOI

      10.1007/978-3-030-76657-3_17

    • ISBN
      9783030766566, 9783030766573
    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Optimal Parenthesizing of Geometric Algebra Products2020

    • Author(s)
      Stephane Breuils, Vincent Nozick, Akihiro Sugimoto
    • Journal Title

      Proc. of Computer Graphics International (CGI2020)

      Volume: LNCS 12221 Pages: 492-500

    • DOI

      10.1007/978-3-030-61864-3_42

    • ISBN
      9783030618636, 9783030618643
    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors2020

    • Author(s)
      Breuils, Stephane, Vincent Nozick, Akihiro Sugimoto
    • Journal Title

      ArXiv abs/2002.11313

      Volume: N/A

    • Related Report
      2020 Annual Research Report
    • Open Access / Int'l Joint Research
  • [Journal Article] A Geometric Algebra Library Generator2019

    • Author(s)
      Breuils, S., Nozick, V., Fuchs, L.
    • Journal Title

      Advances in Applied Clifford Algebras

      Volume: 29 Issue: 4

    • DOI

      10.1007/s00006-019-0987-7

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment2019

    • Author(s)
      Breuils, S., Fuchs, L., Hitzer, E., Nozick, V., Sugimoto, A.
    • Journal Title

      Advances in Applied Clifford Algebras

      Volume: 29 Issue: 3

    • DOI

      10.1007/s00006-019-0974-z

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces2019

    • Author(s)
      Breuils, S., Nozick, V., Fuchs, L., Sugimoto, A.
    • Journal Title

      Proc. of Computer Graphics International (CGI 2019)

      Volume: LNCS 11542 Pages: 523-534

    • DOI

      10.1007/978-3-030-22514-8_52

    • ISBN
      9783030225131, 9783030225148
    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Visiting Bijective Digitized Reflections and Rotations using Geometric Algebra2021

    • Author(s)
      Stephane Breuils, Akihiro Sugimoto, Yukiko Kenmochi
    • Organizer
      IAPR International Conference on Discrete Geometry and Mathematical Morphology (DGMM2021)
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Toward Characterization of 3D Bijective Digitized Reflections2021

    • Author(s)
      Stephane Breuils, Yukiko Kenmochi, Eric Andres, Akihiro Sugimoto
    • Organizer
      Groupe de Travail de Geometrie Discrete et Morphologie Mathematique (GT-GDMM)
    • Related Report
      2021 Annual Research Report
  • [Presentation] Digital geometry and geometric algebra2021

    • Author(s)
      Stephane Breuils, Akihiro Sugimoto, Yukiko Kenmochi
    • Organizer
      Groupe de Travail en Geometrie Discrete (Discrete geometry working group)
    • Related Report
      2020 Annual Research Report
  • [Presentation] Optimal Parenthesizing of Geometric Algebra Products2020

    • Author(s)
      Stephane Breuils, Vincent Nozick, Akihiro Sugimoto
    • Organizer
      Computer Graphics International (CGI2020)
    • Related Report
      2020 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Hands on Geometric Algebra2020

    • Author(s)
      Stephane Breuils, Akihiro Sugimoto
    • Organizer
      Joint 11th International Conference on Soft Computing and Intelligent Systems and 21st International Symposium on Advanced Intelligent Systems (SCIS&ISIS2020)
    • Related Report
      2020 Annual Research Report
    • Invited
  • [Presentation] Bijective digitized reflections and rotations using geometric algebra2020

    • Author(s)
      Stephane Breuils, Akihiro Sugimoto, Yukiko Kenmochi
    • Organizer
      JFLI seminar at NII
    • Related Report
      2020 Annual Research Report
  • [Presentation] Geometric Algebra, Geometric Transformation and some applications in Control Theory2020

    • Author(s)
      Breuils, S.
    • Organizer
      Seminar at Kogakuin University
    • Related Report
      2019 Annual Research Report
  • [Presentation] Geometric Algebra and Computer Graphics2020

    • Author(s)
      Breuils, S.
    • Organizer
      Seminar at Kyushu University
    • Related Report
      2019 Annual Research Report
  • [Presentation] Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces2019

    • Author(s)
      Breuils, S., Nozick, V., Fuchs, L., Sugimoto, A.
    • Organizer
      Computer Graphics International (CGI 2019)
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Geometric Algebra Operators and Application to Quadric Surfaces2019

    • Author(s)
      Breuils, S.
    • Organizer
      Seminar of Japanese-French Laboratory for Informatics
    • Related Report
      2019 Annual Research Report
  • [Presentation] Hands on Geometric Algebra: A Geometry and Computational Unifying Framework for Representing and Transforming Geometric Objects2019

    • Author(s)
      Breuils, S.
    • Organizer
      Tutorial at Asian Conference on Pattern Recognition 2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research

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Published: 2019-07-30   Modified: 2024-03-26  

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