A B-spline basis for C1 quadratic splines on triangulations with a 10-split

The paper considers the macro-element splitting technique that refines every triangle of the initial triangulation into ten smaller triangles. The resulting refinement is an extension of the well-known Powell-Sabin 6-split and enables a construction of polynomial C1 splines of degree two interpolating first order Hermite data at the vertices of the initial triangulation. A particular construction, called a balanced 10-split, is presented that allows a numerically stable B-spline representation of such splines. This amounts to, firstly, defining locally supported basis functions for the macro-element space that form a convex partition of unity, and, secondly, expressing the coefficients of the spline represented in this basis by the means of spline values and derivatives at the vertices of the initial triangulation.