Computation of Normals for Stationary Subdivision Surfaces

摘要

This paper proposes a method for computing of normals for stationary subdivision surfaces. In [1,2], we derived a new necessary and sufficient condition for C~k-continuity of stationary subdivision schemes. First, we showed that tangent plane continuity is equivalent to the convergence of difference vectors. Thus, using "normal subdivision matrix", we derived a necessary and sufficient condition of tangent plane continuity for stationary subdivision at extraordinary points (including degree 6). Moreover, we derived a necessary and sufficient condition for C~1-continuity. Using the analysis, we show that at general points on stationary subdivision surfaces, the computation of the exact normal is an infinite sum of linear combinations of cross products of difference vectors even if the surfaces are C~1-continuous. So, it is not computable. However, we can compute the exact normal of subdivision surfaces at the limit position of a vertex of original mesh or of j-th subdivided mesh for any finite j even if the surfaces are not regular.