Four-point curve subdivision based on iterated chordal and centripetal parameterizations

  • Authors:

  • Nira Dyn;Michael S. Floater;Kai Hormann

  • Affiliations:

  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Centre of Mathematics for Applications, Department of Informatics, University of Oslo, 0316 Oslo, Norway;Department of Informatics, Clausthal University of Technology, 38678 Clausthal-Zellerfeld, Germany

  • Venue:
  • Computer Aided Geometric Design
  • Year:
  • 2009

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Abstract

Dubuc's interpolatory four-point scheme inserts a new point by fitting a cubic polynomial to neighbouring points over uniformly spaced parameter values. In this paper we replace uniform parameter values by chordal and centripetal ones. Since we update the parameterization at each refinement level, both schemes are non-linear. Because of this data-dependent parameterization, the schemes are only invariant under solid body and isotropic scaling transformations, but not under general affine transformations. We prove convergence of the two schemes and bound the distance between the limit curve and the initial control polygon. Numerical examples indicate that the limit curves are smooth and that the centripetal one is tighter, as suggested by the distance bounds. Similar to cubic spline interpolation, the use of centripetal parameter values for highly non-uniform initial data yields better results than the use of uniform or chordal ones.