A method for analysis of C1-continuity of subdivision surfaces

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A sufficient condition for C1-continuity of subdivision surfaces was proposed by Reif [Comput. Aided Geom. Design, 12 (1995), pp. 153-174.] and extended to a more general setting in [D. Zorin, Constr. Approx., accepted for publication]. In both cases, the analysis of C1-continuity is reduced to establishing injectivity and regularity of a characteristic map. In all known proofs of C1-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 C1-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 C1-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.

Original languageEnglish (US)
Pages (from-to)1677-1708
Number of pages32
JournalSIAM Journal on Numerical Analysis
Issue number5
StatePublished - 2000


  • Arbitrary meshes
  • Interval arithmetics
  • Stationary subdivision
  • Subdivision surfaces

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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