Towards the multivariate simplotope spline: continuity conditions in a class of mixed simplotopic grids

TL;DR

This paper introduces a new method for defining continuity conditions for tensor-product Bernstein polynomials on mixed grids, extending existing approaches to more general grid configurations in multivariate spline theory.

Contribution

A novel approach to establish continuity conditions for tensor-product Bernstein polynomials on mixed simplotopic grids, based on defining a surrounding simplex and adapting multivariate spline conditions.

Findings

Two- and three-dimensional results align with existing literature.

Method potentially applicable to more general grid types.

Abstract

Smooth joins of simplex Bernstein-B\'ezier polynomials have been studied extensively in the past. In this paper a new method is proposed to define continuity conditions for tensor-product Bernstein polynomials on a class of mixed grids that meets certain out-of-facet parallelism criteria. The conditions are derived by first defining a simplex around the simplotopic bases of the tensor-product polynomials. Then the continuity conditions in the multivariate simplex spline defined on the resulting simplices, are adapted to hold for the tensor-product polynomials. The two- and three-dimensional results agree with the results found in the literature. It is expected that the method can be employed in more general grids.