This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the "shape" of the set. For that purpose, this paper introduces the formal notion of the family of ff-shapes of a finite point set in R³. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter ff 2 IR controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time O(n²), worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.

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.

In this semi-tutorial paper, a comprehensive survey of closest-point search methods for lattices without a regular structure is presented. The existing search strategies are described in a unified framework, and differences between them are elucidated. An efficient closest-point search algorithm, based on the Schnorr-Euchner variation of the Pohst method, is implemented. Given an arbitrary point x 2 R m and a generator matrix for a lattice , the algorithm computes the point of that is closest to x. The algorithm is shown to be substantially faster than other known methods, by means of a theoretical comparison with the Kannan algorithm and an experimental comparison with the Pohst algorithm and its variants, such as the recent Viterbo-Boutros decoder. The improvement increases with the dimension of the lattice. Modifications of the algorithm are developed to solve a number of related search problems for lattices, such as finding a shortest vector, determining the kissing number, compu...

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The shape of a protein is important for its functions. This includes the location and size of identifiable regions in its complement space. We formally define pockets as regions in the complement with limited accessibility from the outside. Pockets can be efficiently constructed by an algorithm based on alpha complexes. The algorithm is implemented and applied to proteins with known three-dimensional conformations.

A sliver is a tetrahedron whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.

this report, the model was tested on various subsonic and transonic flow problems: flat plates, airfoils, wakes, etc. The model consists of a single advectiondiffusion equation with source term for a field variable which is the product of turbulence Reynolds number and kinematic viscosity, e RT . This variable is proportional to the eddy viscosity except very near a solid wall. The model equation is of the form: D( e RT ) Dt =(c ffl 2 f 2 (y + ) \Gamma c ffl 1 ) q e RT P +( + t oe R )r 2 ( e RT ) \Gamma 1 oe ffl (r t ) \Delta r( e RT ): (6:3:3) In this equation P is the production of turbulent kinetic energy and is related to the mean flow velocity rate-of-strain and the kinematic eddy viscosity t . Equation (6.3.3) depends on distance to solid walls in two ways. First, the damping function f 2 appearing in equation (6.3.3) depends directly on distance to the wall (in wall units). Secondly, t depends on e R t and damping functions which require distance to the wall

Two linear time (and hence asymptotically optimal) algorithms for computing the Euclidean distance transform of a two-dimensional binary image are presented. The algorithms are based on the construction and regular sampling of the Voronoi diagram whose sites consist of the unit (feature) pixels in the image. The first algorithm, which is of primarily theoretical interest, constructs the complete Voronoi diagram. The second, more practical, algorithm constructs the Voronoi diagram where it intersects the horizontal lines passing through the image pixel centres. Extensions to higher dimensional images and to other distance functions are also discussed. 1 Introduction A two-dimensional binary image is a function, I, from the elements of an n by m array, referred to as pixels, to f0; 1g. Pixels of unit (respectively, zero) value are referred to as feature (respectively, background) pixels of the image. We associate the pixel in row r and column c with the Cartesian point (c; r). Thus, an...

1. Multiple polylogarithms. We define them by the power series expansion: (1)