Catmull–Clark subdivision surface

Statement

Let $M$ be a closed polygon mesh in $\R^3$: finitely many vertices, edges and polygonal faces, where every edge lies on exactly two faces. The valence of a vertex is the number of edges at it; on such a mesh it equals the number of faces at it. One step of Catmull–Clark subdivision builds a new mesh $M'$ as follows.

1. For each face $f$, the face point $F_f$ is the average of the vertices of $f$.
2. For each edge $e$ with endpoints $P_1, P_2$ and adjacent faces $f_1, f_2$, the edge point is $E_e = \frac{1}{4}(P_1 + P_2 + F_{f_1} + F_{f_2})$.
3. Each old vertex $P$ of valence $n$ moves to $P' = \frac{1}{n}\big(Q + 2R + (n - 3) P\big)$. Here $Q$ is the average of the face points of the $n$ faces at $P$. Also, $R$ is the average of the midpoints of the $n$ edges at $P$.
4. Each face $f$ with $k$ vertices is replaced by $k$ quadrilaterals. The one at a vertex $P$ of $f$ has as vertices $P'$, the edge points of the two edges of $f$ at $P$, and $F_f$.

Call a vertex extraordinary if its valence is not $4$. Then the following hold.

1. Every face of $M'$ is a quadrilateral, and from the second step on, the number of extraordinary vertices does not change.
2. Consider a part of the mesh where all faces are quadrilaterals and all vertices have valence $4$. There the step is the refinement rule of the uniform bicubic B-spline surface with the mesh as control net. So the repeated meshes converge there to that surface, which is twice continuously differentiable.
3. The repeated meshes $M, M', M'', \ldots$ converge to a limit surface, which has a continuous tangent plane also at the limits of the extraordinary vertices.