Showing papers by "John Keyser published in 1999"

Fast computation of generalized Voronoi diagrams using graphics hardware

Abstract: We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling; I.3.3 [Computer Graphics]: Picture/Image Generation. Additional

Accurate computation of the medial axis of a polyhedron

Abstract: We present an accurate and efficient algorithm to compute the internal Voronoi region and medial axis of a 3-D polyhedron. It uses exact arithmetic and representations for accurate computation of the medial axis. The sheets, seams, and junctions of the medial axis are represented as trimmed quadric surfaces, algebraic space curves, and points with algebraic coordinates, respectively. The algorithm works by recursively finding neighboring junctions along the seam curves. It uses spatial decomposition and linear programming to speed up the search step. We also present a new algorithm for analysis of the topology of an algebraic plane curve, which is the core of our medial axis algorithm. To speed up the computation, we have designed specialized algorithms for fast computation on implicit geometric structures. These include lazy evaluation based on multivariate Stiirm sequences, fast resultant computation, curve topology analysis, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

MAPC: a library for efficient and exact manipulation of algebraic points and curves

Abstract: We present MAPC a library for exact representation of geometric objects speci cally points and algebraic curves in the plane Our library makes use of several new algorithms which we present here including methods for nding the sign of a determinant nding intersections between two curves and breaking a curve into monotonic segments These algorithms are used to speed up the underlying computations The library provides C classes that can be used to easily instantiate manipulate and perform queries on points and curves in the plane The point classes can be used to represent points known in a variety of ways e g as exact rational coordinates or algebraic numbers in a uni ed manner The curve class can be used to represent a portion of an algebraic curve We have used MAPC for applications dealing with algebraic points and curves including sorting points along a curve computing arrangement of curves medial axis computations and boundary evaluation of spline primitives As compared to earlier algorithms and implementations utilizing exact arithmetic our library is able to achieve more than an order of magnitude improvement in performance