Adaptive isotopic approximation of nonsingular curves

The parametrizability and nonlocal isotopy approach

Long Lin, Chee Yap

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local 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 o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

Original languageEnglish (US)
Title of host publicationProceedings of the 25th Annual Symposium on Computational Geometry, SCG'09
Pages351-360
Number of pages10
DOIs
StatePublished - 2009
Event25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark
Duration: Jun 8 2009Jun 10 2009

Other

Other25th Annual Symposium on Computational Geometry, SCG'09
CountryDenmark
CityAarhus
Period6/8/096/10/09

Fingerprint

Isotopy
Curve
Approximation
Subdivision
Subdivision Algorithm
Arbitrary
Intersect
Aspect Ratio
Rectangle
Aspect ratio
Singularity
Computing
Cell
Experimental Results
Geometry

Keywords

  • Curve approximation
  • Exact numerical algorithms
  • Isotopy
  • Meshing
  • Parametrizability
  • Subdivision algorithms
  • Topological correctness

ASJC Scopus subject areas

  • Computational Mathematics
  • Geometry and Topology
  • Theoretical Computer Science

Cite this

Lin, L., & Yap, C. (2009). Adaptive isotopic approximation of nonsingular curves: The parametrizability and nonlocal isotopy approach. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09 (pp. 351-360) https://doi.org/10.1145/1542362.1542423

Adaptive isotopic approximation of nonsingular curves : The parametrizability and nonlocal isotopy approach. / Lin, Long; Yap, Chee.

Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 351-360.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Lin, L & Yap, C 2009, Adaptive isotopic approximation of nonsingular curves: The parametrizability and nonlocal isotopy approach. in Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. pp. 351-360, 25th Annual Symposium on Computational Geometry, SCG'09, Aarhus, Denmark, 6/8/09. https://doi.org/10.1145/1542362.1542423
Lin L, Yap C. Adaptive isotopic approximation of nonsingular curves: The parametrizability and nonlocal isotopy approach. In Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. p. 351-360 https://doi.org/10.1145/1542362.1542423
Lin, Long ; Yap, Chee. / Adaptive isotopic approximation of nonsingular curves : The parametrizability and nonlocal isotopy approach. Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09. 2009. pp. 351-360
@inproceedings{ddc527bda4ff4bc587936d33d97a6e4e,
title = "Adaptive isotopic approximation of nonsingular curves: The parametrizability and nonlocal isotopy approach",
abstract = "We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local 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 o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.",
keywords = "Curve approximation, Exact numerical algorithms, Isotopy, Meshing, Parametrizability, Subdivision algorithms, Topological correctness",
author = "Long Lin and Chee Yap",
year = "2009",
doi = "10.1145/1542362.1542423",
language = "English (US)",
isbn = "9781605585017",
pages = "351--360",
booktitle = "Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09",

}

TY - GEN

T1 - Adaptive isotopic approximation of nonsingular curves

T2 - The parametrizability and nonlocal isotopy approach

AU - Lin, Long

AU - Yap, Chee

PY - 2009

Y1 - 2009

N2 - We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local 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 o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

AB - We consider domain subdivision algorithms for computing isotopic approximations of nonsingular curves represented implicitly by an equation f (X, Y) = 0. Two algorithms in this area are from Snyder (1992) and Plantinga & Veg- ter (2004). We introduce a new algorithm that combines the advantages of these two algorithms: like Snyder, we use the parametrizability criterion for subdivision, and like Plantinga & Vegter we exploit non-local 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 o transversally. Our algorithm is also easy to implement exactly. We report on very encouraging preliminary experimental results, showing that our algorithms can be much more efficient than both Plantinga & Vegter's and Snyder's algorithms.

KW - Curve approximation

KW - Exact numerical algorithms

KW - Isotopy

KW - Meshing

KW - Parametrizability

KW - Subdivision algorithms

KW - Topological correctness

UR - http://www.scopus.com/inward/record.url?scp=70849116923&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70849116923&partnerID=8YFLogxK

U2 - 10.1145/1542362.1542423

DO - 10.1145/1542362.1542423

M3 - Conference contribution

SN - 9781605585017

SP - 351

EP - 360

BT - Proceedings of the 25th Annual Symposium on Computational Geometry, SCG'09

ER -