General Midpoint Subdivision
In this paper, we introduce two generalizations of midpoint subdivision and analyze the smoothness of the resulting subdivision surfaces at regular and extraordinary points. The smoothing operators used in midpoint and mid-edge subdivision connect the midpoints of adjacent faces or of adjacent edges, respectively. An arbitrary combination of these two operators and the refinement operator that splits each face with m vertices into m quadrilateral subfaces forms a general midpoint subdivision operator. We analyze the smoothness of the resulting subdivision surfaces by estimating the norm of a special second order difference scheme and by using established methods for analyzing midpoint subdivision. The surfaces are smooth at their regular points and they are also smooth at extraordinary points for a certain subclass of general midpoint subdivision schemes. Generalizing the smoothing rules of non general midpoint subdivision schemes around extraordinary and regular vertices or faces results in a class of subdivision schemes, which includes the Catmull-Clark algorithm with restricted parameters. We call these subdivision schemes generalized Catmull-Clark schemes and we analyze their smoothness properties.