On the support of recursive subdivision

Abstract

We study the support of subdivision schemes: that is, the region of the subdivision surface that is affected by the displacement of a single control point. Our main results cover the regular case, where the mesh induces a regular Euclidean tesselation of the local parameter space. If <i>n</i> is the ratio of similarity between the tesselations at steps <i>k</i> and <i>k</i> − 1 of the refinement, we show that <i>n</i> determines the extent of this region and largely determines whether its boundary is polygonal or fractal. In particular if <i>n</i> = 2 (or <i>n</i><sup>2</sup> = 2 because we can always take double steps) 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.