Evolutions of polygons in the study of subdivision surfaces.

Abstract

We employ the theory of evolving n-gons in the study of subdivision surfaces. We show that for subdivision schemes with small stencils the eige¬nanalysis of an evolving polygon, corresponding either to a face or to the 1-¬ring neighborhood of a vertex, complements in a geometrically intuitive way the eigenanalysis of the subdivision matrix. In the applications, we study the types of singularities that may appear on a subdivision surface, and we find properties of the subdivision surface that depend on the initial control polyhedron only.