A Hermite Subdivision Scheme for the Evaluation of the Powell-Sabin 12-Split Element

It is observed that the Powell-Sabin 12-split triangle is re nable since the same split of the 4 similar subtriangles of a triangle contains the lines of split of the original triangle. This property of the split is the key to the existence of a subdivision scheme, for the evaluation of the C quadratic spline on the split which interpolates function and gradient values at the 3 vertices of the triangle, and normal derivatives at the midpoints of the edges. Explicit formulae for the Hermite subdivision step are given. For rendering the interpolant it is suggested to use the triangulation and the function values at the vertices obtained after a small number of subdivision iterations, and to use the known values of the gradient at the vertices to obtain the normals to the surface at the vertices of the triangulation. The shading of the 3D triangulation can then be done by Gouraud shading. It is further suggested to perturb the C-Hermite subdivision scheme which evaluates the above interpolant on the Powell-Sabin 12-split triangle, to obtain other C schemes with a shape parameter. x