Analyzing Midpoint Subdivision

TL;DR

This paper proves that midpoint subdivision surfaces of any degree greater than or equal to 2 are C1-continuous at extraordinary points, extending previous analytical results up to degree 9.

Contribution

It develops general analysis tools to establish C1-continuity for midpoint subdivision surfaces of all degrees >= 2.

Findings

Midpoint subdivision of any degree >= 2 is C1-continuous at extraordinary points.

The paper extends previous results from degree 9 to all degrees >= 2.

Provides a unified framework for analyzing midpoint subdivision surfaces.

Abstract

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.