$h$-$\gamma$ Blossoming, $h$-$\gamma$ Bernstein Bases, and $h$-$\gamma$ B\'{e}zier Curves for Translation Invariant $\left(\gamma_{1},\gamma_{2}\right)$ Spaces

TL;DR

This paper introduces a new mathematical framework combining $ ext{h}$-blossoming with $ ext{$oldsymbol{ extgamma}$}$-blossoming for translation invariant function spaces, leading to novel Bernstein bases and Bézier curves with recursive algorithms and properties.

Contribution

It develops the $h$-$ extgamma$ blossom and associated Bernstein bases and Bézier curves for translation invariant $ ext{$oldsymbol{ extgamma}$}$ spaces, unifying and extending existing concepts.

Findings

Derived recursive evaluation algorithms for $h$-$ extgamma$ schemes.

Established subdivision procedures and degree elevation formulas.

Proved properties and identities for the new $h$-$ extgamma$ basis functions.

Abstract

A $\left(\gamma_{1}, \gamma_{2}\right)$ space of order $n$ is a space of univariate functions spanned by $\left\{\gamma_{1}^{n-k}(x), \gamma_{2}^{k}(x)\right\}_{k=0}^{n}$. A $\left(\gamma_{1}, \gamma_{2}\right)$ space is said to be translation invariant if $\gamma_{1}(x-h)$ and $\gamma_{2}(x-h)$ can be expressed as nonsingular linear combinations of $\gamma_{1}(x)$ and $\gamma_{2}(x)$. Translation invariant $\left(\gamma_{1}, \gamma_{2}\right)$ spaces include polynomials $\left(\gamma_{1}(x)=1, \gamma_{2}(x)=x\right)$, trigonometric functions $\left(\gamma_{1}(x)=\cos x, \gamma_{2}(x)=\sin x\right)$, hyperbolic functions $\left(\gamma_{1}(x)=\cosh x, \gamma_{2}(x)=\sinh x\right)$, and their discrete analogues. We merge $\gamma$-blossoming for $\left(\gamma_{1}, \gamma_{2}\right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-B\'{e}zier curves to construct a novel $h$-$\gamma$ blossom for translation invariant $\left(\gamma_{1}, \gamma_{2}\right)$ spaces generated by two continuous, linearly independent functions $\gamma_{1}$ and $\gamma_{2}$. Based on this $h$-$\gamma$ blossom, we define $h$-$\gamma$ Bernstein bases and $h$-$\gamma$ B\'{e}zier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$\gamma$ Bernstein and $h$-$\gamma$ B\'{e}zier schemes.