Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An evaluation algorithm for q-Bézier triangular patches formed by convex combinations

Jorge Delgado, Héctor Orera, Juan Manuel Peña

TL;DR

This work extends the univariate $q$-Bernstein basis to triangular domains and develops a convex-combination–based evaluation framework. It defines triangular $q$-Bernstein polynomials $B_{ijk}^n(u,v)$ using $q$-binomial coefficients, derives two recurrences, proves nonnegativity for $q\in(0,1]$, and establishes a de Casteljau–type evaluation algorithm for exact evaluation. It proves partition of unity, provides a degree-elevation formula, and shows that the degree-$n$ basis spans $\mathbb{P}^n$ with the definition of $q$-Bézier patches; a stability analysis compares conditioning to the classical Bernstein basis, concluding the latter is better conditioned due to a nonnegative change-of-basis matrix. The results deliver a robust triangular $q$-Bézier design framework with explicit algorithms and theoretical guarantees.

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.

An evaluation algorithm for q-Bézier triangular patches formed by convex combinations

TL;DR

This work extends the univariate -Bernstein basis to triangular domains and develops a convex-combination–based evaluation framework. It defines triangular -Bernstein polynomials using -binomial coefficients, derives two recurrences, proves nonnegativity for , and establishes a de Casteljau–type evaluation algorithm for exact evaluation. It proves partition of unity, provides a degree-elevation formula, and shows that the degree- basis spans with the definition of -Bézier patches; a stability analysis compares conditioning to the classical Bernstein basis, concluding the latter is better conditioned due to a nonnegative change-of-basis matrix. The results deliver a robust triangular -Bézier design framework with explicit algorithms and theoretical guarantees.

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.
Paper Structure (7 sections, 8 theorems, 42 equations, 5 figures, 1 algorithm)

This paper contains 7 sections, 8 theorems, 42 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic notations
  3. An extension of the triangular Bernstein polynomials
  4. An evaluation algorithm of the $q$--Bézier representation of polynomials over triangles
  5. $q$--Bernstein basis in a triangle and $q$--Bézier patches
  6. Stabiliy of the basis
  7. Conclusions and future work

Key Result

Proposition 3.1

The triangular $q$--Bernstein polynomials def.qBer are given by and where $B_{000}^0(u,v)=1$ and $B_{ijk}^n(u,v)=0$ for any negative index $i,j,k$.

Figures (5)

  • Figure 1: The effect of changing the q-parameter on the cubic $q$--Bernstein polynomial $B_{003}^3$
  • Figure 2: The effect of changing the q-parameter on the cubic $q$--Bernstein polynomial $B_{012}^3$
  • Figure 3: The effect of changing the q-parameter on the cubic $q$--Bernstein polynomial $B_{004}^4$
  • Figure 4: Cubic q-Bézier patch
  • Figure 5: Cubic q-Bézier patch

Theorems & Definitions (16)

  • Proposition 3.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.3
  • proof
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • ...and 6 more