Technical reports

On the support of recursive subdivision

I.P. Ivrissimtzis, M.A. Sabin, N.A. Dodgson

September 2002, 20 pages

An updated, improved version of this report has been published in ACM Trans. Graphics 23(4):1043–1060, October 2004 [doi:10.1145/1027411.1027417]

DOI: 10.48456/tr-544

Abstract

We study the support of subdivision schemes, that is, the area of the subdivision surface that will be affected by the displacement of a single control point. Our main results cover the regular case, where the mesh induces a regular Euclidean tessellation of the parameter space. If n is the ratio of similarity between the tessellation at step k and step k−1 of the subdivision, we show that this number determines if the support is polygonal or fractal. In particular if n=2, as it is in the most schemes, the support is a polygon whose vertices can be easily determined. If n is not equal to two as, for example, in the square root of three scheme, the support is usually fractal and on its boundary we can identify sets like the classic ternary Cantor set.

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BibTeX record

@TechReport{UCAM-CL-TR-544,
  author =	 {Ivrissimtzis, I.P. and Sabin, M.A. and Dodgson, N.A.},
  title = 	 {{On the support of recursive subdivision}},
  year = 	 2002,
  month = 	 sep,
  url = 	 {https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-544.pdf},
  institution =  {University of Cambridge, Computer Laboratory},
  doi = 	 {10.48456/tr-544},
  number = 	 {UCAM-CL-TR-544}
}