BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CSL-TR-98-764
       ENTRY:: July 21, 1998
ORGANIZATION:: Stanford University, Computer Systems Laboratory
       TITLE:: A Method for Analysis of C(superscript 1)-Continuity of Subdivision Surfaces
        TYPE:: Technical Report
      AUTHOR:: Zorin, Denis
        DATE:: May 1998
       PAGES:: 72
    ABSTRACT:: A sufficient condition for C(superscript 1)-continuity of 
subdivision surfaces was proposed by Rief [17] and extended to a more
general setting in [22].  In both cases, the analysis of 
C(superscript 1)-continuity is reduced to establishing injectivity and
regularity of a characteristic map.  In all known proofs of C(superscript 1)-continuity,
explicit representation of the limit surface on an annular region 
was 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(superscript 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(superscript 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. 
       NOTES:: [Adminitrivia V1/Prg/19980121]
         END:: STAN//CSL-TR-98-764