Gelfond-Bezier Curves

TL;DR

This paper demonstrates that Gelfond-Bernstein bases in Muntz spaces can be derived as limits of Chebyshev-Bernstein bases, enabling simpler curve design algorithms like de Casteljau and blossom.

Contribution

It introduces Gelfond-Bernstein bases as limits of Chebyshev-Bernstein bases and translates key curve design concepts to these bases for improved simplicity.

Findings

Gelfond-Bernstein bases are limits of Chebyshev-Bernstein bases as interval parameter approaches zero.

Curve design concepts like de Casteljau algorithm are adapted to Gelfond-Bernstein bases.

Simpler algorithms are achieved compared to traditional Chebyshev-Bernstein bases.

Abstract

We show that the generalized Bernstein bases in Muntz spaces defined by Hirschman and Widder [7] and extended by Gelfond [6] can be obtained as limits of the Chebyshev-Bernstein bases in Muntz spaces with respect to an interval [a,1] as the 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 translated from the general theory of Chebyshev blossom in Muntz 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.