A correction to the definition of regularity in Chapter 8

The initial discussion of regularity in section 8.2.1 (pp.250) is in  error.  The goal of the discussion is to estabilish conditions on the  characteristic map to have a globally-defined inverse.  This inverse map is then used to reduce the smoothness analysis at extraordinary vertices  to the functional case using the function .  The error arises from the non-standard definition of regularity given  in the second paragraph on page 250. In the text, is erroneously defined to be "regular" if is   and onto.  The correct definition is that is regular at  a point if   has continuous first derivatives and satisfies the Jacobian condition of  equation 8.16 at .

Under this corrected definition, Reif [128] shows that a sufficient  condition for to exist is that is regular and injective ().  In practice, proving injectivity of directly can be difficult.  Instead, Chapter 8 takes an approach similar  to that of Zorin [below] in establishing the existence of .  The basic idea is to show that is regular on the annulus and that the image of this annulus under winds around the origin exactly once.  The text describes a computational  test for regularity in section 8.3.3 while Zorin's paper describes a method  for computing the winding number of .
      

CITATION

D. Zorin, "A method for analysis of -continuity of subdivision surfaces," Siam Journal of Numerical Analysis, Vol.37, No.5, pp.1677-1708, 2000.