Preconditions on geometrically sensitive subdivision schemes

Our objective is to create subdivision schemes with limit surfaces which are surfaces useful in engineering (spheres, cylinders, cones etc.) without resorting to special cases. The basic idea explored by Sabin et al. [1] in the curve case is that if the property that all vertices lie on an object of the required class can be preserved through the subdivision refinement, it will be preserved into the limit surface also. The next obvious step was to try a bivariate example. We therefore identified the simplest possible scheme and implemented it. However, this misbehaved quite dramatically. This note, by doing the limit analysis (which should, in hindsight, have preceded the implementation), identifies why the misbehaviour occurred, and draws conclusions about how the problems should be avoided. 1 The simplest geometry sensitive bivariate scheme Sabin et al. describe an interpolating curve scheme which preserves circles [1]. We therefore sought to find an interpolating surface scheme which preserves spheres. We wanted to make the experiment as simple as possible. We therefore chose that the new facets should be defined by √ 3 rules [2], so that each new facet joins one original vertex to new face-vertices in two adjacent triangles (Figure 1). This means that the only new vertices to be positioned are ones logically in the middles of the old facets: because the scheme is interpolating there are also new ones which are simply positioned exactly at the old ones. These decisions meant that only triangulations had a well defined refinement, not general polyhedra, and we also chose to limit this experiment to polyhedra without boundary. This was quite acceptable for a first experiment. Our first idea was that each new face-vertex in a triangulation should be chosen to make the estimator of mean curvature there equal to the mean of the mean curvatures at the three vertices of the face. [email protected] [email protected]