$h$-$γ$ Blossoming, $h$-$γ$ Bernstein Bases, and $h$-$γ$ Bézier Curves for Translation Invariant $\left(γ_{1},γ_{2}\right)$ Spaces
This work extends existing blossoming theory to a broader class of function spaces, offering a systematic approach for curve design in non-polynomial spaces.
The paper develops a unified framework for h-γ Bernstein bases and Bézier curves in translation invariant (γ1, γ2) spaces, providing recursive evaluation, subdivision, degree elevation, and interpolation algorithms.
A $\left(γ_{1}, γ_{2}\right)$ space of order $n$ is a space of univariate functions spanned by $\left\{γ_{1}^{n-k}(x), γ_{2}^{k}(x)\right\}_{k=0}^{n}$. A $\left(γ_{1}, γ_{2}\right)$ space is said to be translation invariant if $γ_{1}(x-h)$ and $γ_{2}(x-h)$ can be expressed as nonsingular linear combinations of $γ_{1}(x)$ and $γ_{2}(x)$. Translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces include polynomials $\left(γ_{1}(x)=1, γ_{2}(x)=x\right)$, trigonometric functions $\left(γ_{1}(x)=\cos x, γ_{2}(x)=\sin x\right)$, hyperbolic functions $\left(γ_{1}(x)=\cosh x, γ_{2}(x)=\sinh x\right)$, and their discrete analogues. We merge $γ$-blossoming for $\left(γ_{1}, γ_{2}\right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-Bézier curves to construct a novel $h$-$γ$ blossom for translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces generated by two continuous, linearly independent functions $γ_{1}$ and $γ_{2}$. Based on this $h$-$γ$ blossom, we define $h$-$γ$ Bernstein bases and $h$-$γ$ Bézier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$γ$ Bernstein and $h$-$γ$ Bézier schemes.