Computation of Normals for Stationary Subdivision Surfaces

  • Hiroshi Kawaharada
  • Kokichi Sugihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


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 C k -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 C 1-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 C 1-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.


Subdivision Scheme Subdivision Surface Original Mesh Limit Position Jordan Normal Form 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hiroshi Kawaharada
    • 1
  • Kokichi Sugihara
    • 1
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of Tokyo 

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