Construction of $C^2$ cubic splines on arbitrary triangulations

TL;DR

This paper introduces a method for constructing stable, high-quality $C^2$ cubic spline functions on arbitrary triangulations by employing Wang-Shi macro-structures and simplex spline bases, enabling local construction and easy visualization.

Contribution

It develops a new $C^2$ cubic spline space on arbitrary triangulations with a stable basis, local construction, and explicit simplex spline basis functions, advancing spline theory and applications.

Findings

Stable dimension and optimal approximation power of the spline space.

Explicit simplex spline basis with recurrence, differentiation, and partition of unity.

Facilitates local construction and visualization of splines on complex triangulations.

Abstract

In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in such space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of $C^2$ cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for $C^2$ joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang-Shi macro-structure is transparent to the user. Stable global bases for the full space of $C^2$ cubics on the Wang-Shi refined triangulation $\mathcal{T}$ are deduced from the local simplex spline basis by extending the concept of minimal determining sets.