Midpoint subdivision generalizes the Lane-Riesenfeld algorithm for uniform tensor product splines and can also be applied to non regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo-Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull-Clark algorithm. In 2001, Zorin and Schroeder were able to prove C1-continuity for midpoint subdivision surfaces analytically up to degree 9. Here, we develop general analysis tools to show that the limiting surfaces under midpoint subdivision of any degree >= 2 are C1-continuous at their extraordinary points.

The paper was improved by adding more explanations and by adding an illustration of how the statements depend on each other. We combined a few theorems to simplify the structure of the paper and better described the meaning of the statements and how they fit into the overall proof. 24 pages, 10 figures