C1 spline implicitization of planar curves

Mohamed Shalaby; Bert Jüttler; Josef Schicho

doi:10.1007/978-3-540-24616-9_10

C¹ spline implicitization of planar curves

Mohamed Shalaby, Bert Jüttler, Josef Schicho

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

4 Scopus citations

Abstract

We present a new method for constructing a low degree C¹ implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The spline segments are implicitized. After multiplying the implicit spline segments by suitable polynomial factors the resulting bivariate functions are joined along suitable transversal lines. This yields a globally C¹ bivariate function.

Original language	English (US)
Title of host publication	Automated Deduction in Geometry
Editors	Franz Winkler
Publisher	Springer Verlag
Pages	161-177
Number of pages	17
ISBN (Print)	3540209271, 9783540209270
DOIs	https://doi.org/10.1007/978-3-540-24616-9_10
State	Published - 2004
Externally published	Yes
Event	4th International Workshop on Automated Deduction in Geometry, ADG 2002 - Hagenberg Castle, Austria Duration: Sep 4 2002 → Sep 6 2002

Publication series

Name	Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science)
Volume	2930
ISSN (Print)	0302-9743

Other

Other	4th International Workshop on Automated Deduction in Geometry, ADG 2002
Country/Territory	Austria
City	Hagenberg Castle
Period	09/4/02 → 09/6/02

Keywords

Approximation
B-spline
Implicitization
Knot removal

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-540-24616-9_10

Cite this

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title = "C1 spline implicitization of planar curves",

abstract = "We present a new method for constructing a low degree C1 implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The spline segments are implicitized. After multiplying the implicit spline segments by suitable polynomial factors the resulting bivariate functions are joined along suitable transversal lines. This yields a globally C1 bivariate function.",

keywords = "Approximation, B-spline, Implicitization, Knot removal",

author = "Mohamed Shalaby and Bert J{\"u}ttler and Josef Schicho",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2004.; 4th International Workshop on Automated Deduction in Geometry, ADG 2002 ; Conference date: 04-09-2002 Through 06-09-2002",

year = "2004",

doi = "10.1007/978-3-540-24616-9_10",

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series = "Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science)",

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booktitle = "Automated Deduction in Geometry",

address = "Germany",

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Shalaby, M, Jüttler, B & Schicho, J 2004, C¹ spline implicitization of planar curves. in F Winkler (ed.), Automated Deduction in Geometry. Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science), vol. 2930, Springer Verlag, pp. 161-177, 4th International Workshop on Automated Deduction in Geometry, ADG 2002, Hagenberg Castle, Austria, 09/4/02. https://doi.org/10.1007/978-3-540-24616-9_10

C¹ spline implicitization of planar curves. / Shalaby, Mohamed; Jüttler, Bert; Schicho, Josef.
Automated Deduction in Geometry. ed. / Franz Winkler. Springer Verlag, 2004. p. 161-177 (Lecture Notes in Artificial Intelligence (Subseries of Lecture Notes in Computer Science); Vol. 2930).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We present a new method for constructing a low degree C1 implicit spline representation of a given parametric planar curve. To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces. Adaptive knot removal, which is based on spline wavelets, is used to reduce the number of segments. The spline segments are implicitized. After multiplying the implicit spline segments by suitable polynomial factors the resulting bivariate functions are joined along suitable transversal lines. This yields a globally C1 bivariate function.

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