Tim Culver

Fast computation of generalized Voronoi diagrams using graphics hardware

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

Interactive motion planning using hardware-accelerated computation of generalized Voronoi diagrams

We present techniques for fast motion planning by using discrete approximations of generalized Voronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a 3D distance mesh for each Voronoi site. The sites can be points, line segments, polygons, polyhedra, curves and surfaces. The computation of the generalized Voronoi diagram provides fast proximity query toolkits for motion planning. The tools provide the distance to the nearest obstacle stored in the Z-buffer, as well as the Voronoi boundaries, Voronoi vertices and weighted Voronoi graphs extracted from the frame buffer using continuation methods. We have implemented these algorithms and demonstrated their performance for path planning in a complex dynamic environment composed of more than 140,000 polygons.

Accurate computation of the medial axis of a polyhedron

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.

Exact computation of the medial axis of a polyhedron

We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding neighboring junctions along the seam curves. To speed up the computation, we have designed specialized algorithms for fast computation with algebraic curves and surfaces. These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples.

ESOLID¿a system for exact boundary evaluation

Abstract We present a system, ESOLID, that performs exact boundary evaluation of low-degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating-point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating-point filters, arbitrary floating-point arithmetic with error bounds, and lower-dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating-point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating-point boundary evaluation system on most cases.

Efficient and exact manipulation of algebraic points and curves

Abstract An important part of solid modeling systems based on curved primitives is the representation and manipulation of algebraic plane curves with rational coefficients and points with algebraic coordinates. These objects are often approximated by floating-point numbers and spline curves, which are easy to manipulate, but are subject to accuracy and robustness problems. Exact computation can eliminate these numerical robustness problems, but efficient exact methods have not been available. Moreover, it is widely believe that exact arithmetic is impractical for manipulating curved primitives. In this paper, we present an efficient approach for exact manipulation of algebraic points and non-singular curves in the plane. We describe the underlying representations and discuss techniques for making exact computations more efficient through two algorithms and the use of floating-point speedups. Specifically, we describe algorithms for curve–curve intersections and curve topology. We also discuss various issues related to their implementation in a library, MAPC. We demonstrate their performance on a number of applications including curve topology evaluation, computing arrangements of curves and boundary evaluation of low degree algebraic solids.

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

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

BOOLE: A BOUNDARY EVALUATION SYSTEM FOR BOOLEAN COMBINATIONS OF SCULPTURED SOLIDS

In this paper we describe a system, BOOLE, that generates the boundary representations (B-reps) of solids given as a CSG expression in the form of trimmed Bezier patches. The system makes use of techniques from computational geometry, numerical linear algebra and symbolic computation to generate the B-reps. Given two solids, the system first computes the intersection curve between the two solids using our surface intersection algorithm. Using the topological information of each solid, it computes various components within each solid generated by the intersection curve and their connectivity. The component classification step is performed by ray-shooting. Depending on the Boolean operation performed, appropriate components are put together to obtain the final solid. We also present techniques to parallelize this system on shared memory multiprocessor machines. The system has been successfully used to generate B-reps for a number of large industrial models including parts of a notional submarine storage and handling room (courtesy - Electric Boat Inc.) and Bradley fighting vehicle (courtesy - Army Research Labs). Each of these models is composed of over 8000 Boolean operations and is represented using over 50,000 trimmed Bezier patches. Our exact representation of the intersection curve and use of stable numerical algorithms facilitate an accurate boundary evaluation at every Boolean set operation and generation of topologically consistent solids.

PRECISE: efficient multiprecision evaluation of algebraic roots and predicates for reliable geometric computation

Many geometric problems like generalized Voronoi diagrams, medial axis computations and boundary evaluation involve computation and manipulation of non-linear algebraic primitives like curves and surfaces. The algorithms designed for these problems make decisions based on signs of geometric predicates or on the roots of polynomials characterizing the problem. The reliability of the algorithm depends on the accurate evaluation of these signs and roots. In this paper, we present a {\em naive precision-driven computational model} to perform these computations reliably and demonstrate its effectiveness on a certain class of problems like sign of determinants with rational entries, boundary evaluation and curve arrangements. We also present a novel algorithm to compute all the roots of a univariate polynomial to any desired accuracy. The computational model along with the underlying number representation, precision-driven arithmetic and all the algorithms are implemented as part of a stand-alone software library, PRECISE.

ESOLID---A System for Exact Boundary Evaluation

We present a system, ESOLID, that performs exact boundary evaluation of low degree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRL-CAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixed-precision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases.