A Hermite Subdivision Scheme for PS - 12 3 Initialization 1 . subdivision step

It is observed that the Powell-Sabin 12-split triangle is reen-able 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 1 quadratic spline on the split which interpolates function and gradient values at the 3 vertices of the triangle, and normal derivatives at the mid-points 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 triangu-lation. The shading of the 3D triangulation can then be done by Gouraud shading. It is further suggested to perturb the C 1-Hermite subdivision scheme which evaluates the above interpolant on the Powell-Sabin 12-split triangle, to obtain other C 1 schemes with a shape parameter. x1. Introduction For bivariate smooth spline spaces of low degree on triangulations it is diicult to construct basis functions with local support. One approach which leads to good results is the splitting of each triangle into subtriangles according to the same rule of splitting. Among the known splits is the Powell-Sabin 12-split (PS-12 split). Here each triangle is divided into 12 subtriangles by connecting each vertex of the triangle to the midpoint of the opposite edge and connecting the midpoints, see Fig. 1. On this split there is a unique quadratic C 1-spline interpolant to function values and gradients at the vertices and cross derivatives at the midpoints of the three edges. ((9]). This interpolant is called the PS-12 split element. Due to the large number of subtriangles in this split it is hard to compute these elements, (but see 1]). Yet this split has the following advantage ((8,7]): ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.