Circle-preserving schemes

It was shown, in chapter 21.2 above, that with a stationary scheme nonzero artifacts can be achieved after the first refinement at a finite number of imposed spatial frequencies. However, at the second refinement the original spatial frequency will appear to have halved, relative to the density of the 

With a non-stationary scheme, however, for a given initial spatial frequency of polygon, (i.e. a fixed known number of vertices per complete cycle, or vertices forming a regular polygon), the coefficients can vary at each step so that the halving relative frequency of the signal is tracked by a zero of the kernel. 

This tracking is regular enough that the updating of the coefficients can be made into a regular recurrence. 

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The circle-preserving schemes described above are essentially just carrying out this recipe. 

M.A.Sabin, and N.A.Dodgson A circle-preserving variant of the four-point subdivision scheme. pp275-286 in Mathematical Methods for Curves and Surfaces, (eds Daehlen, Morken and Schumaker), Nash-boro Press 2005 ISBN 0-9728482-4-X... 

L.Romani A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control. CAGD 27(1), pp36-47, 2010 C.Deng and G.Wang Incenter subdivision scheme for curve interpolation. CAGD 27(1), pp48-59, 2010... 

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