Polygonal decomposition of the 1-ring neighborhood of the Catmull-Clark scheme

We propose a polygonal decomposition of the 1-ring neighborhood of a quadrilateral mesh, which is suitable for the study of the Catmull-Clark subdivision scheme. The initial configuration consists of 2n planar 2n-gons and under the Catmull-Clark subdivision they transform into 4n planar n-gons coming in pairs of coplanar polygons and quadruples of parallel polygons. We calculate the eigenvalues and eigenvectors of the transformations of these configurations showing their relation with the tangent plane and the curvature properties of the subdivision surface. Using direct computations on circulant-block matrices, we show how the same eigenvalues can be analytically deduced from the subdivision matrix.