A Method for Analysis of C1 -Continuity of Subdivision Surfaces

A sufficient condition for C 1-continuity of subdivision surfaces was proposed by Reif [17] and extended to a more general setting in [22]. In both cases, the analysis of C 1-continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C 1-continuity, explicit representation of the limit surface on an annular region was used to establish regularity, and a variety of relatively complex techniques were used to establish injectivity. We propose a new approach to this problem: we show that for a general class of subdivision schemes, regularity can be inferred from the properties of a sufficiently close linear approximation, and injectivity can be verified by computing the index of a curve. An additional advantage of our approach is that it allows us to prove C 1-continuity for all valences of vertices, rather than for an arbitrarily large, but finite number of valences. As an application, we use our method to analyze C 1-continuity of most stationary subdivision schemes known to us, including interpolating Butterfly and Modified Butterfly schemes, as well as the Kobbelt's interpolating scheme for quadrilateral meshes. 1 Introduction Subdivision is becoming increasingly popular as a surface representation in computer graphics applications. To ensure that a subdivision algorithm has the desired behavior for almost all input data, a theoretical analysis of the surface has to be performed. For subdivision on arbitrary meshes, even the analysis of the basic property of the surfaces, C 1-continuity, poses a considerable challenge; [18, 9, 14, 15, 19]. In this paper we describe a set of theoretical results and algorithms that make it possible to perform the C 1-continuity tests automatically. The principal result allowing one to analyze C 1-continuity of most subdivision schemes, is the sufficient condition of Reif [17]. This condition reduces the analysis of stationary subdivision to the analysis of a single map, called the characteristic map, for each valence of vertices in the mesh. The analysis of C 1-continuity is performed in three steps for each valence: 1. compute the control net of the characteristic map; 2. prove that the characteristic map is regular; 3. prove that the characteristic map is injective. This map can be expressed in a closed form for spline-based subdivision schemes, such as Loop, Catmull-Clark and Doo-Sabin. For these schemes, proving regularity of the characteristic map is tedious but straightforward, as the Jacobian of the map can be …