# Computer Laboratory

Technical reports

# Subdivision as a sequence of sampled Cp surfaces and conditions for tuning schemes

Cédric Gérot, Loïc Barthe, Neil A. Dodgson, Malcolm A. Sabin

March 2004, 68 pages

## Abstract

We deal with practical conditions for tuning a subdivision scheme in order to control its artifacts in the vicinity of a mark point. To do so, we look for good behaviour of the limit vertices rather than good mathematical properties of the limit surface. The good behaviour of the limit vertices is characterised with the definition of C2-convergence of a scheme. We propose necessary explicit conditions for C2-convergence of a scheme in the vicinity of any mark point being a vertex of valency n or the centre of an n-sided face with n greater or equal to three. These necessary conditions concern the eigenvalues and eigenvectors of subdivision matrices in the frequency domain. The components of these matrices may be complex. If we could guarantee that they were real, this would simplify numerical analysis of the eigenstructure of the matrices, especially in the context of scheme tuning where we manipulate symbolic terms. In this paper we show that an appropriate choice of the parameter space combined with a substitution of vertices lets us transform these matrices into pure real ones. The substitution consists in replacing some vertices by linear combinations of themselves. Finally, we explain how to derive conditions on the eigenelements of the real matrices which are necessary for the C2-convergence of the scheme.

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## BibTeX record

```@TechReport{UCAM-CL-TR-583,
author =	 {G{\'e}rot, C{\'e}dric and Barthe, Lo{\"\i}c and Dodgson,
Neil A. and Sabin, Malcolm A.},
title = 	 {{Subdivision as a sequence of sampled Cp surfaces and
conditions for tuning schemes}},
year = 	 2004,
month = 	 mar,
url = 	 {http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-583.pdf},
institution =  {University of Cambridge, Computer Laboratory},
number = 	 {UCAM-CL-TR-583}
}
```