A simple matrix form for degree reduction of Bézier curves using Chebyshev-Bernstein basis transformations

Abedallah Rababah; Byung Gook Lee; Jaechil Yoo

doi:10.1016/j.amc.2006.01.034

A simple matrix form for degree reduction of Bézier curves using Chebyshev-Bernstein basis transformations

Abedallah Rababah, Byung Gook Lee, Jaechil Yoo

Research output: Contribution to journal › Article › peer-review

30 Citations (Scopus)

Abstract

We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of Bézier curves with respect to the weighted L₂-norm for the interval [0, 1], using the weight function w (x) = 1 / sqrt(4 x - 4 x²). The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.

Original language	English
Pages (from-to)	310-318
Number of pages	9
Journal	Applied Mathematics and Computation
Volume	181
Issue number	1
DOIs	https://doi.org/10.1016/j.amc.2006.01.034
Publication status	Published - Oct 1 2006
Externally published	Yes

Keywords

Basis transformations
Bézier curves
Chebyshev polynomials
Continuity conditions
Degree elevation
Degree reduction

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2006.01.034

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title = "A simple matrix form for degree reduction of B{\'e}zier curves using Chebyshev-Bernstein basis transformations",

abstract = "We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of B{\'e}zier curves with respect to the weighted L2-norm for the interval [0, 1], using the weight function w (x) = 1 / sqrt(4 x - 4 x2). The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.",

keywords = "Basis transformations, B{\'e}zier curves, Chebyshev polynomials, Continuity conditions, Degree elevation, Degree reduction",

author = "Abedallah Rababah and Lee, {Byung Gook} and Jaechil Yoo",

note = "Funding Information: Lee was supported by Korea Science and Engineering Foundation Grant (R05-2004-000-10968-0) and Yoo was supported by Dongeui University.",

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doi = "10.1016/j.amc.2006.01.034",

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AU - Rababah, Abedallah

AU - Lee, Byung Gook

AU - Yoo, Jaechil

N1 - Funding Information: Lee was supported by Korea Science and Engineering Foundation Grant (R05-2004-000-10968-0) and Yoo was supported by Dongeui University.

PY - 2006/10/1

Y1 - 2006/10/1

N2 - We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of Bézier curves with respect to the weighted L2-norm for the interval [0, 1], using the weight function w (x) = 1 / sqrt(4 x - 4 x2). The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.

AB - We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of Bézier curves with respect to the weighted L2-norm for the interval [0, 1], using the weight function w (x) = 1 / sqrt(4 x - 4 x2). The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.

KW - Basis transformations

KW - Bézier curves

KW - Chebyshev polynomials

KW - Continuity conditions

KW - Degree elevation

KW - Degree reduction

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A simple matrix form for degree reduction of Bézier curves using Chebyshev-Bernstein basis transformations

Abstract

Keywords

ASJC Scopus subject areas

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