Advances in Computational Mathematics 14: 157–174, 2001.  2001 Kluwer Academic Publishers. Printed in the Netherlands. Curvature integrability of subdivision surfaces Ulrich Reif a and Peter Schröder b a Technische Universität Darmstadt, FB Mathematik, AG3, Schloßgartenstr. 7, D-64289 Darmstadt, Germany E-mail: reif@mathematik.tu-darmstadt.de b Caltech, MS 256-80, 1200 E. California Boulevard, Pasadena, CA 91125, USA E-mail: ps@cs.caltech.edu Received 2 February 2000; revised 9 September 2000; accepted 15 November 2000 Communicated by C. Micchelli We examine the smoothness properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their L p class based on the eigen- structure of the subdivision matrix. As a result we can show that the popular Loop and Catmull–Clark schemes (among many others) have square integrable principal curvatures en- abling their use as shape functions in FEM treatments of the thin shell equations. Keywords: subdivision, smoothness, function spaces, approximation, thin shell equations AMS subject classification: 65D17, 65N30 1. Introduction Subdivision surfaces are a popular modeling primitive in computer graphics and computer assisted geometric design. They offer many advantages in applications, chief among them their ability to model smooth surfaces of arbitrary topology. Figure 1 shows examples of surfaces based on the scheme of Loop [13] (a) and Catmull–Clark [3] (b). Historically subdivision schemes were developed as generalizations of uniform B-spline (midpoint) knot insertion algorithms [5] to settings with topologically irreg- ular control meshes, i.e., those with vertices whose valence is other than four. Doo and Sabin [6] generalized bi-quadratic splines, while Catmull and Clark [3] generalized bi- cubic splines in this way. Similarly, Loop [13] generalized quartic box splines (in this case irregular vertices are those with valence other than six) and Habib and Warren [8] as well as Peters and Reif [14] generalized the quadratic 4-direction spline. The idea of sub- division was also studied independent of the spline setting. For example, Dyn et al. [7] (and later Zorin et al. [25]) described an interpolating scheme based on triangles, while Leber [12] and Kobbelt [11] developed interpolating schemes based on quadrilaterals. In all these settings smooth surfaces are created through a limit process of recursive refinement of a given initial polyhedron. For example, in the Loop scheme an initial