A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
Stationary Subdivision
Subdivision Methods for Geometric Design: A Constructive Approach
Subdivision Methods for Geometric Design: A Constructive Approach
Stationary subdivision and multiresolution surface representations
Stationary subdivision and multiresolution surface representations
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This paper proposes a method for computing of normals for stationary subdivision surfaces. In [1, 2], we derived a new necessary and sufficient condition for Ck-continuity of stationary subdivision schemes. First, we showed that tangent plane continuity is equivalent to the convergence of difference vectors. Thus, using “normal subdivision matrix” [3], we derived a necessary and sufficient condition of tangent plane continuity for stationary subdivision at extraordinary points (including degree 6). Moreover, we derived a necessary and sufficient condition for C1-continuity. Using the analysis, we show that at general points on stationary subdivision surfaces, the computation of the exact normal is an infinite sum of linear combinations of cross products of difference vectors even if the surfaces are C1-continuous. So, it is not computable. However, we can compute the exact normal of subdivision surfaces at the limit position of a vertex of original mesh or of j-th subdivided mesh for any finite j even if the surfaces are not regular.