Toward a curvature-continuous stationary subdivision algorithm

We generalize the Loop subdivision algorithm for deriving a surface from a triangulated control mesh. Under subdivision, a vertex in the control mesh tends toward a limit point on the surface. Viewing this algorithm as an iterative smoothing at the vertices in the mesh, we identify the stochastic matrix which describes the refinement of the two- neighborhood of a vertex. By a unique choice of two parameters, the eigenstructure of this matrix produces a quadratic Taylor expansion with remainder for a family of curves coming into the limit point. The constant term in this expansion yields the limit point; the linear terms yield the tangent space to the surface at the limit point. Any curvature in the surface at the limit point is determined by the quadratic terms in this Taylor expansion.