6. Concluding remarks
In this paper we have presented a new C1 cubic spline space defined over a triangulation endowed with a PS-refinement. Thanks to the locally imposed C2 super-smoothness, the proposed PS3-spline space has a simple dimension formula, namely 3nv +2ne , and the space is a close extension of the classical CT3-spline space. In addition, we have constructed a normalized B-spline basis for this space. The basis functions have a local support, they are nonnegative, and they form a partition of unity. We have also described how to compute from the control points of a PS3-spline its corresponding Bézier control net in a stable way. In the literature one finds few other normalized B-spline representations for C1 cubic splines on triangulations with a macro-structure. For example, such a representation exists for RCT3-splines (Speleers, 2010b) and for cubic PS-splines with a different super-smoothness (Lamnii et al., 2014). For the sake of convenience, the latter splines will be referred to as PS3-splines in the following. The proposed new cubic B-spline representation has some favorable properties with respect to the other ones.
