Ringing: Frugal subdivision of curves and surfaces (IEEE-Computer Graphics and Applications) (Jarek Rossignac and Abhishek Venkatesh):
Abstract: 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 C2 curves, and Maillot and Stam’s generalization of these, which was recently analyzed by Rossignac and Schaefer and may be used to produce C4 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)2d+5 vertices. Rendering each span independently reduces temporary storage requirement (footprint) to 2d+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.
Keywords: Subdivision, Refinements, B-splines, Curves, Surfaces, Animation, GPU, Graphics, Streaming.
3D-Visualization of Power System Data Using Triangulation and Subdivision techniques. (Abhishek Venkatesh, George Cokkinides, A. P. Sakis Meliopoulos)
Abstract: 3-D surface visualizations of various power system operating quantities has always been challenging in terms of correctly capturing the changes of an arbitrary geographical shape power system. Triangulation methods offer promise for meaningful 3-D surface visualizations of such systems. In this paper we propose a scheme for such visualizations based on subdivision of triangle meshes. Input consists of various power system quantities such as voltage magnitude, voltage phase, reactive power flow, real power flow, electric current, etc. The data may be available from simulations or from real time streaming data from a model that is two-dimensional (geographic). We first perform a Delaunay Triangulation on the set of 2D sites and generate a triangle mesh. This triangle mesh is used to represent a coarse 3D surface. The height of this surface at a site is equal to the power system quantity at that site. This surface is refined using the butterfly subdivision scheme with an additional constraint that the heights of the interpolated vertices lie within the bounds of the original vertices from which they were interpolated. After each level of subdivision, we perform a modified Laplacian smoothing to compensate for the discontinuity introduced due to this bounding. The method is suitable for effective visualization of large geographic data. Example visualizations and performance indices are provided in the paper.
Keywords: Delaunay triangulation, corner table, butterfly subdivision, laplacian smoothing, 3D-vizualization, interpolating, surface.