Publication details for Dr Ioannis IvrissimtzisIvrissimtzis, Ioannis, Sabin, Malcolm & Dodgson, Neil (2004). On the support of recursive subdivision. ACM transactions on graphics 23(4): 1043-1060.
- Publication type: Journal Article
- ISSN/ISBN: 0730-0301
- DOI: 10.1145/1027411.1027417
- Keywords: Cantor set, subdivision, support.
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
We study the support of subdivision schemes: that is, the region of the subdivision surface, which is 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 local parameter space. If n is the ratio of similarity between the tessellations at steps k and k-1 of the refinement, we show that n determines the extent of this region and largely determines whether its boundary is polygonal or fractal. In particular, if n = 2 the support is a convex polygon whose vertices can easily be determined. In other cases, whether the boundary of the support is fractal or not depends on whether there are sufficient points with non-zero coefficients in the edges of the convex hull of the mask. If there are enough points on every such edge, the support is again a convex polygon. If some edges have enough points and others do not, the boundary can consist of a fractal assembly of an unbounded number of line segments.
ALEXA, M. 2002. Refinement operators for triangle meshes. Comput. Aided Geom. Des. 19(3), 169–172.
CATMULL, E. AND CLARK, J. 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided
Des. 10, 350–355.
DESLAURIERS, G. AND DUBUC, S. 1989. Symmetric Iterative Interpolation Processes. Constr. Approx. 5, 49–68.
DODGSON, N. A., IVRISSIMTZIS, I. P., AND SABIN, M. A. 2003. Characteristics of dual √3 subdivision schemes. In Curve and Surface
Fitting: Saint-Malo 2002. Nashboro Press, Brentwood, TN, 119–128.
DODGSON, N. A., SABIN, M. A., BARTHE, L., AND HASSAN, M. F. 2002. Towards a ternary interpolating subdivision
scheme for the triangular mesh. Tech. Rep. 539, University of Cambridge Computer Laboratory. July 2002.
DOO, D. AND SABIN,M. 1978. Behaviour of recursive division surfaces near extraordinary points. Comput. Aid. Des. 10, 356–360.
DYN, N., LEVIN, D., AND GREGORY, J. A. 1990. A butterfly subdivision scheme for surface interpolation with tension control. ACM
Trans. Graph. 9(2), 160–169.
IVRISSIMITZIS, I. P., DODGSON, N. A., AND SABIN, M. A. 2004. A generative classification of subdivision schemes with lattice
transformations. Comput. Aided Geom. Des. 22(1), 99–109.
KANNAN, V. 1994. Cantor set: From classical to modern. Math. Stud. 63, 1-4, 243–257.
KOBBELT, L. 2000. √3-Subdivision. In SIGGRAPH 2000 Conference Proceedings. ACM, New York, 103–112.
LOOP, C. T. 1987. Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics.
PETERS, J. AND REIF, U. 1997. The simplest subdivision scheme for smoothing polyhedra. ACM Trans. Graph. 16(4), 420–431.
SABIN, M. A. AND BARTHE, L. 2003. Artifacts in recursive subdivision schemes. In Curve and Surface Fitting: Saint-Malo 2002.
Nashboro Press, Brentwood, TN, 353–362.
VELHO, L. AND ZORIN, D. 2001. 4–8 subdivision. Comput. Aided Geom. Des. 18, 397–427.