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

31 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

Access to Document

10.1016/j.amc.2006.01.034

Cite this

@article{c015eacbe9aa4d17b2992375253f4d6d,

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.",

year = "2006",

month = oct,

day = "1",

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

language = "English",

volume = "181",

pages = "310--318",

journal = "Applied Mathematics and Computation",

issn = "0096-3003",

publisher = "Elsevier Inc.",

number = "1",

}

TY - JOUR

T1 - A simple matrix form for degree reduction of Bézier curves using Chebyshev-Bernstein basis transformations

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

UR - http://www.scopus.com/inward/record.url?scp=33749519804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749519804&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2006.01.034

DO - 10.1016/j.amc.2006.01.034

M3 - Article

AN - SCOPUS:33749519804

SN - 0096-3003

VL - 181

SP - 310

EP - 318

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this