Algorithms and Identities for B$\acute{e}$zier curves via Post Quantum Blossom

TL;DR

This paper introduces a post quantum calculus-based blossom for Bézier curves, leading to new identities, formulas, and recursive algorithms that enhance the mathematical framework and computational methods for Bernstein bases.

Contribution

It develops a novel post quantum blossom framework, providing new identities and recursive algorithms for Bézier curves and Bernstein bases.

Findings

New post quantum identities for Bernstein bases

Recursive evaluation algorithms for Bézier curves

Post quantum Marsden's identity

Abstract

In this paper, a new analogue of blossom based on post quantum calculus is introduced. The post quantum blossom has been adapted for developing identities and algorithms for Bernstein bases and B$\acute{e}$zier curves. By applying the post quantum blossom, various new identities and formulae expressing the monomials in terms of the post quantun Bernstein basis functions and a post quantun variant of Marsden's identity are investigated. For each post quantum B$\acute{e}$zier curves of degree $m,$ a collection of $m!$ new, affine invariant, recursive evaluation algorithms are derived.