Construction of $$C^2$$ Cubic Splines on Arbitrary Triangulations

Abstract

AbstractIn this paper, we address the problem of constructing $$C^2$$ C 2 cubic spline functions on a given arbitrary triangulation $${\mathcal {T}}$$ T . To this end, we endow every triangle of $${\mathcal {T}}$$ T with a Wang–Shi macro-structure. The $$C^2$$ C 2 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 $$C^2$$ 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 $$C^2$$ 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 $$C^2$$ C 2 cubics on the Wang–Shi refined triangulation $${\mathcal {T}}$$ T are deduced from the local simplex spline basis by extending the concept of minimal determining sets.