On the support of recursive subdivision

Ivrissimtzis, Ioannis; Sabin, Malcolm; Dodgson, Neil

Abstract

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.

Citation

Ivrissimtzis, I., Sabin, M., & Dodgson, N. (2004). On the support of recursive subdivision. ACM Transactions on Graphics, 23(4), 1043-1060. https://doi.org/10.1145/1027411.1027417

• Journal Article Type: Article
• Publication Date: 2004-10
• Deposit Date: Jan 23, 2009
• Journal: ACM Transactions on Graphics
• Print ISSN: 0730-0301
• Publisher: Association for Computing Machinery (ACM)
• Peer Reviewed: Peer Reviewed
• Volume: 23
• Issue: 4
• Pages: 1043-1060
• DOI: https://doi.org/10.1145/1027411.1027417
• Keywords: Cantor set.