An evaluation algorithm for q-B\'ezier triangular patches formed by convex combinations

TL;DR

This paper introduces an extension of q-Bernstein basis functions for triangular domains, establishing their properties, basis formation, and a linear convex combination evaluation algorithm, enhancing computational techniques for q-Bézier patches.

Contribution

It presents a new extension of q-Bernstein basis functions for triangles, along with recurrence relations, basis proof, and a de Casteljau type evaluation algorithm.

Findings

Basis functions form a polynomial space on triangles

Recurrence relations are established for the basis functions

A linear convex combination evaluation algorithm is developed

Abstract

An extension to triangular domains of the univariate q-Bernstein basis functions is introduced and analyzed. Some recurrence relations and properties such as partition of unity and degree elevation are proved for them. It is also proved that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. In addition, it is presented a de Casteljau type evaluation algorithm whose steps are all linear convex combinations.