Construction and analysis of cubic Powell-Sabin B-splines

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.