Ringing subdivision of curves and surfaces

Split&Tweak subdivisions iteratively refine a polygon by inserting a vertex in the middle of each edge (Split) and then moving each vertex to an affine combination of five consecutive vertices (Tweak). Special cases include Dyn, Gregory, and Levin’s fourpoint subdivision, Lane and Riesenfeld’s cubic and quintic B-spline subdivision, Rossignac’s subdivision which produces C curves, and Maillot and Stam’s generalization of these, which was recently analyzed by Rossignac and Schaefer and may be used to produce C curves that offer local control and closely approximate the control vertices or minimize the disparity between consecutive refinements. Applying d steps of Split&Tweak subdivision to a control polygon of n vertices requires temporary storage space for (n–5)2+5 vertices. Rendering each span independently reduces temporary storage requirement (footprint) to 2+5 vertices, but increases computation. The ringing approach introduced here reduces the footprint to 4d vertices. We describe an efficient implementation, show applications to surfaces and animations, and report timings comparing CPU and GPU implementations of ringing with the global and per-span approaches.