Adaptive Isotopic Approximation of Nonsingular Curves

The Parameterizability and Nonlocal Isotopy Approach

Long Lin, Chee Yap

Research output: Contribution to journalArticle

Abstract

We consider domain subdivision algorithms for computing isotopic approximations of a nonsingular algebraic curve. The curve is given by a polynomial equation f(X, Y)= 0. Two algorithms in this area are from Snyder (1992) SIGGRAPH Comput. Graphics, 26(2), 121 and Plantinga and Vegter (2004) In Proc. Eurographics Symposium on Geometry Processing, pp. 245-254. We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parameterizability criterion for subdivision, and like Plantinga and Vegter, we exploit nonlocal isotopy. We further extend our algorithm in two important and practical directions: first, we allow subdivision cells to be rectangles with arbitrary but bounded aspect ratios. Second, we extend the input domains to be regions R 0 with arbitrary geometry and which might not be simply connected. Our algorithm halts as long as the curve has no singularities in the region, and intersects the boundary of R 0 transversally. Our algorithm is practical and easy to implement exactly. We report some very encouraging experimental results, showing that our algorithms can be much more efficient than the algorithms of Plantinga-Vegter and Snyder.

Original languageEnglish (US)
Pages (from-to)760-795
Number of pages36
JournalDiscrete and Computational Geometry
Volume45
Issue number4
DOIs
StatePublished - Jun 2011

Fingerprint

Isotopy
Curve
Approximation
Subdivision
Subdivision Algorithm
Polynomial equation
Algebraic curve
Arbitrary
Geometry
Intersect
Aspect Ratio
Rectangle
Aspect ratio
Singularity
Polynomials
Computing
Cell
Experimental Results

Keywords

  • Curve approximation
  • Exact algorithms
  • Isotopy
  • Meshing
  • Parameterizability
  • Subdivision algorithms
  • Topological correctness

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

Adaptive Isotopic Approximation of Nonsingular Curves : The Parameterizability and Nonlocal Isotopy Approach. / Lin, Long; Yap, Chee.

In: Discrete and Computational Geometry, Vol. 45, No. 4, 06.2011, p. 760-795.

Research output: Contribution to journalArticle

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