Fast Computation of Polynomial Data Points Over Simplicial Face Values

Abstract

Polynomial functions [Formula: see text] of degree [Formula: see text] have a form in the Bernstein basis defined over [Formula: see text]-dimensional simplex [Formula: see text]. The Bernstein coefficients exhibit a number of special properties. The function [Formula: see text] can be optimised by the smallest and largest Bernstein coefficients (enclosure bounds) over [Formula: see text]. By a proper choice of barycentric subdivision steps of [Formula: see text], we prove the inclusion property of Bernstein enclosure bounds. To this end, we provide an algorithm that computes the Bernstein coefficients over subsimplices. These coefficients are collected in an [Formula: see text]-dimensional array in the field of computer-aided geometric design. Such a construct is typically classified as a patch. We show that the Bernstein coefficients of [Formula: see text] over the faces of a simplex coincide with the coefficients contained in the patch.