$h$-$\gamma$ Blossoming, $h$-$\gamma$ Bernstein Bases, and $h$-$\gamma$ Bézier Curves for Translation Invariant $\left({\gamma }_1,{\gamma }_2\right)$ Spaces

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 $h$-Bézier 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 $h$-$\gamma$ Bézier 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 $h$-$\gamma$ Bézier schemes.