Gelfond–Bézier curves

Rachid Ait-Haddou; Yusuke Sakane; Taishin Nomura

doi:10.1016/j.cagd.2012.10.002

Gelfond–Bézier curves

Rachid Ait-Haddou, Yusuke Sakane, Taishin Nomura

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

15 Scopus citations

Abstract

We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases in Müntz spaces with respect to an interval [a,1][a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.

Original language	English (US)
Pages (from-to)	199-225
Number of pages	27
Journal	Computer Aided Geometric Design
Volume	30
Issue number	2
DOIs	https://doi.org/10.1016/j.cagd.2012.10.002
State	Published - Feb 2013
Externally published	Yes

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10.1016/j.cagd.2012.10.002

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title = "Gelfond–B{\'e}zier curves",

abstract = "We show that the generalized Bernstein bases in M{\"u}ntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases in M{\"u}ntz spaces with respect to an interval [a,1][a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in M{\"u}ntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.",

author = "Rachid Ait-Haddou and Yusuke Sakane and Taishin Nomura",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: This work was partially supported by the MEXT Global COE project at Osaka University, Japan.",

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T1 - Gelfond–Bézier curves

AU - Ait-Haddou, Rachid

AU - Sakane, Yusuke

AU - Nomura, Taishin

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: This work was partially supported by the MEXT Global COE project at Osaka University, Japan.

PY - 2013/2

Y1 - 2013/2

N2 - We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases in Müntz spaces with respect to an interval [a,1][a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.

AB - We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases in Müntz spaces with respect to an interval [a,1][a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.

UR - http://hdl.handle.net/10754/577065

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DO - 10.1016/j.cagd.2012.10.002

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VL - 30

SP - 199

EP - 225

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

IS - 2

ER -

Gelfond–Bézier curves

Abstract

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