C1 spline implicitization of planar curves

Mohamed Shalaby; Bert Jüttler; Josef Schicho

C¹ spline implicitization of planar curves

Mohamed Shalaby, Bert Jüttler, Josef Schicho

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

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
Publisher	Springer Verlag
Pages	161-177
Number of pages	17
Volume	2930
ISBN (Print)	3540209271, 9783540209270
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

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
Computer Science(all)

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",

year = "2004",

language = "English (US)",

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volume = "2930",

pages = "161--177",

booktitle = "Automated Deduction in Geometry",

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note = "4th International Workshop on Automated Deduction in Geometry, ADG 2002 ; Conference date: 04-09-2002 Through 06-09-2002",

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T1 - C1 spline implicitization of planar curves

AU - Shalaby, Mohamed

AU - Jüttler, Bert

AU - Schicho, Josef

PY - 2004

Y1 - 2004

N2 - 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.

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.

KW - Approximation

KW - B-spline

KW - Implicitization

KW - Knot removal

UR - http://www.scopus.com/inward/record.url?scp=7444243348&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:7444243348

SN - 3540209271

SN - 9783540209270

VL - 2930

SP - 161

EP - 177

BT - Automated Deduction in Geometry

PB - Springer Verlag

T2 - 4th International Workshop on Automated Deduction in Geometry, ADG 2002

Y2 - 4 September 2002 through 6 September 2002

ER -