Constructing Explicit B-Spline

TL;DR

This paper presents an explicit algorithm for constructing multivariate B-spline bases with specified continuity across tetrahedral meshes, facilitating precise geometric modeling.

Contribution

It introduces a direct, analytical method to construct multivariate explicit B-spline bases ensuring $C^{r}$ continuity on complex geometries.

Findings

Provides an explicit construction algorithm for B-splines

Ensures $C^{r}$ continuity across tetrahedral boundaries

Enables precise geometric modeling with B-splines

Abstract

We introduce here a direct method to construct multivariate explicit B-spline bases. B-splines are piecewise polynomials, which are defined on adjacent tetrahedra and which are $C^{r}$ continuous throughout. The $C^{r}$ continuity is enforced by making sure that all directional derivatives of order $r$, and lower, on the boundaries of adjacent tetrahedra give the same values for both tetrahedra. The method presented here is explicit, in that we will provide an algorithm with which one can analytically construct the B-spline base that enforces $C^{r}$ continuity for a given geometry.