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A Comparative Study between the Conventional Euclidean Approach and the Totally 4-Dimensional Approach

Research Project

Project/Area Number 12680400
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Intelligent informatics
Research InstitutionWaseda University

Principal Investigator

YAMAGUCHI Fujio  School of Science and Engineering, Professor, 理工学部, 教授 (50117298)

Co-Investigator(Kenkyū-buntansha) YOSHIDA Norimasa  Dept. of Engineering, Tokyo University of Agriculture and Technology, Research Associate, 工学部, 助手 (70277846)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2002: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥400,000 (Direct Cost: ¥400,000)
KeywordsCAD / CAGD / geometric modeling / computational geometry / 4-D processing / 完全4次元同次処理 / 同次座標 / プリェッカー座標 / ソリッドモデリング / 曲線と曲面 / 同次処理
Research Abstract

A present CAD system is considered to have some deficiencies. Namely, it has many elements that are inaccurate or non-robust, and that the system itself has become too complex. These problems have a direct bearing on a system's reliability. Over the author's course of study, he has come to realize that the major cause of such inherent problems of Euclidean Geometric Processing (EGP) lies in performing division operations, and thus he proposes "Totally Four-dimensional Geometric Processing (TFGP)," which enables us to dispense with the detrimental operations.
The present research is a theoretical and experimental comparison between EGP and TFGP.
(1') Exactness: It can be said that TFGP can perform exact computations as long as it deals with rational numbers. EGP, on the other hand, is obliged to deal with approximated data which are the results of division operations inherent in EGP. This was confirmed through various experimental results.
(2') Robustness: In TFGP, there is no such instabi … More lity as division by zero, because a division operation is not ordinarily performed except at the very end of whole process. While the geometric Newton-Raphson method in EGP occasionally fails when it is applied to rational polynomial curves, i.e., the parameter value diverges and finally halts the algorithm, the TFGP method does not show such non-robustness because it treats a homogeneous curve which is expressed as an ordinary curve of dimension higher by one. This superiority is borne out by many experimental data.
(3') Compactness: A Euclidean geometry tends to be more complicated because it is considered to be a cut of homogeneous one (i.e., linear subspace) of which the dimension is higher by one than that of its counterpart. The increase of geometric types in EGP makes it much more complex. Each type must be mathematically represented individually, and the increased numbers of the combinations must be processed.
As seen above, TFGP is superior to EGP in the above three items and also in terms of generality, unifiability and duality. Less

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report

Research Products

(23 results)

All Other

All Publications (23 results)

  • [Publications] 山内俊哉: "浮動小数点演算ユニットを利用した4×4行列式の適応的符号判定処理"精密工学会誌. 66. 1190-1194 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 山内俊哉: "多変数Sturm列を利用した曲線・曲面の交点の存在判定"精密工学会誌. 67. 498-503 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 山口富士夫: "立体復元CADによる三面図教育"図学研究. 35. 33-34 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 木村雅紀: "同次パラメータ同次幾何的ニュートン法に関する考察"精密工学会誌. 67. 1950-1955 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 山内俊哉: "同次処理に基づく整数演算を用いた多面体ソリッドモデラー"精密工学会誌. (掲載決定済).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshiya Yamauchi: "Efficient Method of Adaptive Sign Detection for 4×4 Determinants Using a Standard Processing Unit"The Visual Computer. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Fujio Yamaguchi: "Computer-Aided Geometric Design"Springer-Verlag. 630 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshiya Yamaguchi: "Adaptive Sign Detection Method of 4×4 Determinants Using a Floating Point Processing Unit"Journal of the Japan Society for Precision Engineering. 66-8. 1190-1194 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshiya Yamauchi: "The Existence Evaluation of Intersection points in Curve/Surface Intersection using Multivariate Sturm Sequences"Journal of the Japan Society for Precision Engineering. 67-3. 498-503 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Fujio Yamaguchi: "A Training of Three-Orthographic View Drawings by Using a Solid Reconstruction CAD system"Journal of Graphic Science of Japan. 35-3. 33-34 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Masanori Kimura: "Study of Homogeneous Parameter, Homogeneous Geometric Newton Method"Journal of the Japan Society for Precision Engineering. 67-12. 1950-1955 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshiya Yamauchi: "A Polyhedral Solid Modeling System Using Exact Arithmetic Based on Homogeneous Processing"Journal of the Japan Society for Precision Engineering. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Toshiya Yamauchi: "Efficient Method of Adaptive Sign Detection for 4×4 Determinants using a Standard Processing Unit"The Visual Computer. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Fujio Yamaguchi: "Computer-Aided Geometric Design"Springer-Verlag. 630 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 木村雅紀: "同次パラメータ同次幾何的ニュートン法に関する考察"精密工学会誌. 67・12. 1950-1955 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山口富士夫: "立体復元CADによる三面図教育"図学研究. 35・3. 33-34 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山内俊哉: "多変数Sturm列を利用した曲線・曲面の交点の存在判定"精密工学会誌. 67・3. 498-503 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山内俊哉: "浮動小数点ユニットを利用した4×4行列式の適応的符号判定処理"精密工学会誌. 66・8. 1190-1195 (2000)

    • Related Report
      2002 Annual Research Report
  • [Publications] Fujio Yamaguchi: "Computer Aided Geometric Design -A Totally Four-Dimensional Approach"Springer-Verlag. 664 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 木村雅紀, 山口富士夫, 渡辺良夫: "同次パラメータ同次幾何的ニュートン法に関する考察"精密工学会誌. 67・12. 1950-1955 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 山口富士夫: "立体復元CADによる三面図教育"図学研究. 35・3. 33-34 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 山内俊哉,山口富士夫: "浮動小数点演算ユニットを利用した4×4行列式の適応的符号判定処理"精密工学会誌. 66巻8号. 1190-1194 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 山内俊哉,山口富士夫: "多変数Sturm列を利用した曲線・曲面の交点の存在判定"精密工学会誌. 67・3(発表予定). (2001)

    • Related Report
      2000 Annual Research Report

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Published: 2000-03-31   Modified: 2016-04-21  

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