Interrogation of differential geometry properties for design and manufacture

Takashi Maekawa, Nicholas M. Patrikalakis

Research output: Contribution to journalArticle

59 Citations (Scopus)

Abstract

This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.

Original languageEnglish
Pages (from-to)216-237
Number of pages22
JournalThe Visual Computer
Volume10
Issue number4
DOIs
Publication statusPublished - 1994 Apr 1
Externally publishedYes

Fingerprint

Polynomials
Geometry
Surface analysis
Nonlinear equations
Machining

Keywords

  • CAD
  • CAGD
  • CAM
  • Curvature analysis
  • Nonlinear equations
  • Rounded interval arithmetic
  • Subdivision

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computer Graphics and Computer-Aided Design

Cite this

Interrogation of differential geometry properties for design and manufacture. / Maekawa, Takashi; Patrikalakis, Nicholas M.

In: The Visual Computer, Vol. 10, No. 4, 01.04.1994, p. 216-237.

Research output: Contribution to journalArticle

Maekawa, Takashi ; Patrikalakis, Nicholas M. / Interrogation of differential geometry properties for design and manufacture. In: The Visual Computer. 1994 ; Vol. 10, No. 4. pp. 216-237.
@article{469eec6f724444a899b4d37cb04eb1c5,
title = "Interrogation of differential geometry properties for design and manufacture",
abstract = "This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.",
keywords = "CAD, CAGD, CAM, Curvature analysis, Nonlinear equations, Rounded interval arithmetic, Subdivision",
author = "Takashi Maekawa and Patrikalakis, {Nicholas M.}",
year = "1994",
month = "4",
day = "1",
doi = "10.1007/BF01901288",
language = "English",
volume = "10",
pages = "216--237",
journal = "Visual Computer",
issn = "0178-2789",
publisher = "Springer Verlag",
number = "4",

}

TY - JOUR

T1 - Interrogation of differential geometry properties for design and manufacture

AU - Maekawa, Takashi

AU - Patrikalakis, Nicholas M.

PY - 1994/4/1

Y1 - 1994/4/1

N2 - This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.

AB - This paper describes a new robust method to decompose a free-form surface into regions with specific range of curvature and provide important tools for surface analysis, tool-path generation, and tool-size selection for numerically controlled machining, tessellation of trimmed patches for surface interrogation and finite-element meshing, and fairing of free-form surfaces. The key element in these techniques is the computation of all real roots within a finite box of systems of nonlinear equations involving polynomials and square roots of polynomials. The free-form surfaces are bivariate polynomial functions, but the analytical expressions of their principal curvatures involve polynomials and square roots of polynomials. Key components are the reduction of the problems into solutions of systems of polynomial equations of higher dimensionality through the introduction of auxiliary variables and the use of rounded interval arithmetic in the context of Bernstein subdivision to enhance the robustness of floating-point implementation. Examples are given that illustrate our techniques.

KW - CAD

KW - CAGD

KW - CAM

KW - Curvature analysis

KW - Nonlinear equations

KW - Rounded interval arithmetic

KW - Subdivision

UR - http://www.scopus.com/inward/record.url?scp=0028259873&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0028259873&partnerID=8YFLogxK

U2 - 10.1007/BF01901288

DO - 10.1007/BF01901288

M3 - Article

AN - SCOPUS:0028259873

VL - 10

SP - 216

EP - 237

JO - Visual Computer

JF - Visual Computer

SN - 0178-2789

IS - 4

ER -