On Regularity of Catmull-Clark Subdivision Surfaces

The regularity of the Catmull-Clark（C-C） subdivision surfaces is studied in this paper, aiming to deduce a simple and easy to use sufficient condition for discriminating the regularity of a C-C subdivision surface. Specifically, we first present the definitions of three types of difference vectors on the mesh, i.e., forward difference vector, central difference vector, and backward difference vector; and then, develop the subdivision formats of the difference vectors; moreover, the relationship between the tangent vectors of a C-C subdivision surface and the deference vectors of its initial control mesh is established by eigen analysis; finally, a sufficient condition for the regularity of a C-C subdivision surface is deduced. Because the condition is represented as the geometric relationship between the difference vectors on the initial control mesh, it has clear geometric meanings. Experiments show that the sufficient condition is easy to validate.