Construction and analysis of cubic Powell–Sabin B-splines

Abstract

We consider a new B-spline representation for the space of C1 cubic splines defined on a triangulation with a Powell–Sabin refinement. The construction is based on lifting particular triangles and line segments from the domain. We prove that the B-splines form a locally supported stable basis and a convex partition of unity. Furthermore, we provide explicit expressions for the B-spline coefficients of any element of the cubic spline space and show how to compute the Bernstein–Bézier form of such a spline in a stable way. The B-spline representation induces a natural control structure that is useful for geometric modelling. Finally, we explore how classical C1 quadratic Powell–Sabin splines and C1 cubic Clough–Tocher splines can be expressed in the new B-spline representation.