C1C1 cubic splines on Powell–Sabin triangulations

A bivariate C1C1 cubic spline space on a triangulation with Powell–Sabin refinement which extends the well-known C1C1 quadratic spline space and has a nested structure is introduced. A construction of a locally supported basis that forms a partition of unity is presented based on choosing particular triangles and line segments in the domain. Further, it is shown how these objects can be determined in order to obtain nonnegative basis functions under a natural restriction on the Powell–Sabin refinement. Geometrically intuitive B-spline representation is proposed which makes these splines a useful tool for CAGD applications.