Isotopic arrangement of simple curves: An exact numerical approach based on subdivision

Jyh Ming Lien, Vikram Sharma, Gert Vegter, Chee Yap

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

Abstract

We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.

Original languageEnglish (US)
Title of host publicationMathematical Software, ICMS 2014 - 4th International Congress, Proceedings
PublisherSpringer Verlag
Pages277-282
Number of pages6
Volume8592 LNCS
ISBN (Print)9783662441985
DOIs
StatePublished - 2014
Event4th International Congress on Mathematical Software, ICMS 2014 - Seoul, Korea, Republic of
Duration: Aug 5 2014Aug 9 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8592 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other4th International Congress on Mathematical Software, ICMS 2014
CountryKorea, Republic of
CitySeoul
Period8/5/148/9/14

Fingerprint

Subdivision
Arrangement
Curve
Approximation algorithms
Derivatives
Curve Approximation
Subdivision Algorithm
Interval Methods
Zero set
Partial derivative
Intersect
Approximation Algorithms
Analytic function
Intersection
Integrate
Generalise
Interval
Computing
Approximation

Keywords

  • arrangement of curves
  • interval arithmetic
  • Isotopy
  • marching-cube
  • subdivision algorithms

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Lien, J. M., Sharma, V., Vegter, G., & Yap, C. (2014). Isotopic arrangement of simple curves: An exact numerical approach based on subdivision. In Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings (Vol. 8592 LNCS, pp. 277-282). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8592 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-662-44199-2_43

Isotopic arrangement of simple curves : An exact numerical approach based on subdivision. / Lien, Jyh Ming; Sharma, Vikram; Vegter, Gert; Yap, Chee.

Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings. Vol. 8592 LNCS Springer Verlag, 2014. p. 277-282 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8592 LNCS).

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

Lien, JM, Sharma, V, Vegter, G & Yap, C 2014, Isotopic arrangement of simple curves: An exact numerical approach based on subdivision. in Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings. vol. 8592 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8592 LNCS, Springer Verlag, pp. 277-282, 4th International Congress on Mathematical Software, ICMS 2014, Seoul, Korea, Republic of, 8/5/14. https://doi.org/10.1007/978-3-662-44199-2_43
Lien JM, Sharma V, Vegter G, Yap C. Isotopic arrangement of simple curves: An exact numerical approach based on subdivision. In Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings. Vol. 8592 LNCS. Springer Verlag. 2014. p. 277-282. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-662-44199-2_43
Lien, Jyh Ming ; Sharma, Vikram ; Vegter, Gert ; Yap, Chee. / Isotopic arrangement of simple curves : An exact numerical approach based on subdivision. Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings. Vol. 8592 LNCS Springer Verlag, 2014. pp. 277-282 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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