Construction of C2 Cubic Splines on Arbitrary Triangulations

Abstract

In this paper, we address the problem of constructing C2 cubic spline functions on a given arbitrary triangulation T. To this end, we endow every triangle of T with a Wang–Shi macro-structure. The C2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this 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 C2 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 C2 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 C2 cubics on the Wang–Shi refined triangulation T are deduced from the local simplex spline basis by extending the concept of minimal determining sets.