Numerische Mathematik

, Volume 113, Issue 2, pp 163–180 | Cite as

Smoothness equivalence properties of univariate subdivision schemes and their projection analogues

  • Philipp GrohsEmail author


We study the following modification of a linear subdivision scheme S: let M be a surface embedded in Euclidean space, and P a smooth projection mapping onto M. Then the P-projection analogue of S is defined as T := PS. As it turns out, the smoothness of the scheme T is always at least as high as the smoothness of the underlying scheme S or the smoothness of P minus 1, whichever is lower. To prove this we use the method of proximity as introduced by Wallner et al. (Constr Approx 24(3):289–318, 2006; Comput Aided Geom Design 22(7):593–622, 2005). While smoothness equivalence results are already available for interpolatory schemes S, this is the first result that confirms smoothness equivalence properties of arbitrary order for general non-interpolatory schemes.

Mathematics Subject Classification (2000)

41AXX 41A25 53B 22E 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.TU GrazInstitute of GeometryGrazAustria

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