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Communicated by F. BrezziWe establish several fundamental properties of analysis-suitable T-splines which are important for design and analysis. First, we characterize T-spline spaces and prove that the space of smooth bicubic polynomials, defined over the extended T-mesh of an analysis-suitable T-spline, is contained in the corresponding analysis-suitable T-spline space. This is accomplished through the theory of perturbed analysis-suitable T-spline spaces and a simple topological dimension formula. Second, we establish the theory of analysis-suitable local refinement and describe the conditions under which two analysissuitable T-spline spaces are nested. Last, we demonstrate that these results can be used to establish basic approximation results which are critical for analysis.

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This paper develops new refinement rules for non-uniform Catmull-Clark surfaces that produce G 1 extraordinary points whose blending functions have a single local maximum. The method consists of designing an "eigen polyhedron" in R 2 for each extraordinary point, and formulating refinement rules for which refinement of the eigen polyhedron reduces to a scale and translation. These refinement rules, when applied to a non-uniform Catmull-Clark control mesh in R 3 , yield a G 1 extraordinary point.