# Gaussian and mean curvature of subdivision surfaces

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- Department of Computer and Information Science and Engineering, University of Florida
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## Abstract

Gaussian and Mean curvature of subdivision surfaces � UFL, CISE TR 2000-01 Jo¨rg Peters ��� , Georg Umlauf ��� March 10, 2000 Abstract By explicitly deriving the curvature of subdivision surfaces in the extraordinary points, we give an alternative, more direct account of the criteria necessary and sufficient for achieving curvature continuity than earlier approaches that locally parametrize the surface by eigenfunctions. The approach allows us to rederive and thus survey the important lower bound results on piecewise polynomial subdivision surfaces by Prautzsch, Reif, Sabin and Zorin, as well as explain the beauty of curvature continuous constructions like Prautzsch’s. The parametrization neutral perspective gives also additional insights into the inherent constraints and stiffness of subdivision surfaces. 1 Introduction Almost all subdivision algorithms in the current literature achieve tangent continuity but not curvature conti- nuity. We give a simple characterization of the causes underlying this phenomenon by explicitly expressing Gaussian and mean curvature in the minimally smooth extraordinary points. This allows us to rederive and thereby survey the important lower bound results of [Sabin ’91, Reif ’96, Zorin ’98, Prautzsch & Reif ’99] and constructions for curvature continuous piecewise polynomial subdivision algorithms by [Prautzsch ’97, Prautzsch & Umlauf ’98b, Reif ’98b]. Beyond this we get additional insights into the inherent constraints and stiffness of such subdivision algorithms. Since a subdivision surface consists of an infinite collection of polynomial pieces around every extraordinary point one might expect such surfaces to be more flexible than spline surfaces. However, we will see that the infinite application of the same subdivision rule enforces strict rules on the piecewise polynomial rings converging towards extraordinary points. For example, the Jacobian of the subdominant eigenfunctions of a curvature continuous subdivision algorithm must have lower degree th

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