We present an efficient O(n + 1/ε^4.5)-time algorithm for computing a (1 + 1/ε)-approximation of the minimum-volume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online [Har00].

Given a set of points P = {p 1 , p 2 ,..., p n } in three dimensions, the width of P, W(P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n + I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and in the worst case I = W(n 2 ). For convex polyhedra, the time complexity becomes O(n + I). If P is a set of points in the plane, the complexity can be reduced to O(n log n). Finally, for simple polygons linear time suffices. Index Terms - Algorithms, antipodal pairs, artificial intelligence, computational geometry, convex hull, geometric complexity, geometric transforms, image processing, minimax approximating line, minimax approximating plane, pattern recognition, rotating calipers, width. 1. Introduction The width of a set of points P (or a simple polygon P) in two dimensions is the minimum distance between parallel lines of support of P (or P). In three d...

In this paper, we consider Dijkstra's algorithm for the single source single target shortest paths problem in large sparse graphs. The goal is to reduce the response time for online queries by using precomputed information. For the result of the preprocessing, we admit at most linear space. We assume that a layout of the graph is given. From this layout, in the preprocessing, we determine for each edge a geometric object containing all nodes that can be reached on a shortest path starting with that edge. Based on these geometric objects, the search space for online computation can be reduced significantly. We present an extensive experimental study comparing the impact of different types of objects. The test data we use are traffic networks, the typical field of application for this scenario.

In this paper we propose several methods that can greatly improve image quality when using the shadow mapping algorithm. Shadow artifacts introduced by shadow mapping are mainly due to low resolution shadow maps and/or the limited numerical precision used when performing the shadow test. These problems especially arise when the light source’s viewing frustum, from which the shadow map is generated, is not adjusted to the actual camera view. We show how a tight fitting frustum can be computed such that the shadow mapping algorithm concentrates on the visible parts of the scene and takes advantage of nearly the full available precision. Furthermore, we recommend uniformly spaced depth values in contrast to perspectively spaced depths in order to equally sample the scene seen from the light source. 1.

Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678. - 2 - ...

This paper presents a rendering method for translucent objects, in which view point and illumination can be modified at interactive rates. In a preprocessing step the impulse response to incoming light impinging at each surface point is computed and stored in two different ways: The local effect on close-by surface points is modeled as a per-texel filter kernel that is applied to a texture map representing the incident illumination. The global response (i.e. light shining through the object) is stored as vertex-to-vertex throughput factors for the triangle mesh of the object. During rendering, the illumination map for the object is computed according to the current lighting situation and then filtered by the precomputed kernels. The illumination map is also used to derive the incident illumination on the vertices which is distributed via the vertex-to-vertex throughput factors to the other vertices. The final image is obtained by combining the local and global response. We demonstrate the performance of our method for several models.

We present an approximation algorithm for computing the diameter of a point-set in d-dimensions. The new algorithm is sensitive to the \hardness" of computing the diameter of the given input, and for most inputs it is able to compute the exact diameter extremely fast. The new algorithm is simple, robust, has good empirical performance, and can be implemented quickly. As such, it seems to be the algorithm of choice in practice for computing/approximating the diameter.

Given a set S of n points in the plane, a quadrangulation of S is a planar subdivision whose vertices are the points of S, whose outer face is the convex hull of S, and every face of the subdivision (except possibly the outer face) is a quadrilateral. We show that S admits a quadrangulation if and only if S does not have an odd number of extreme points. If S admits a quadrangulation, we present an algorithm that computes a quadrangulation of S in O(n log n) time even in the presence of collinear points. If S does not admit a quadrangulation, then our algorithm can quadrangulate S with the addition of one extra point, which is optimal. We also provide an\Omega (n log n) time lower bound for the problem. Finally, our results imply that a k-angulation of a set of points can be achieved with the addition of at most k \Gamma 3 extra points within the same time bound.

this paper we briefly survey the most recent results in the area of facility location, concentrating on versions of the problem that are likely to be unfamiliar to the transportation and management science community and we explore the interaction between facility location problems and the field of computational geometry. Such versions of the problem include the standard models of points as customers and facilities but with geodesic rather than the traditional Minkowski metrics as measures of distance, as well as more complicated models of customers and facilities such as

A fundamental approach in finding efficiently best routes or optimal itineraries in traffic information systems is to reduce the search space (part of graph visited) of the most commonly used shortest path routine (Dijkstra’s algorithm) on a suitably defined graph. We investigate reduction of the search space while simultaneously retaining data structures, created during a preprocessing phase, of size linear (i.e., optimal) to the size of the graph. We show that the search space of Dijkstra’s algorithm can be significantly reduced by extracting geometric information from a given layout of the graph and by encapsulating precomputed shortest path information in resulted geometric objects (containers). We present an extensive experimental study comparing the impact of different types of geometric containers using test data from real-world traffic networks. We also present new algorithms as well as an empirical study for the dynamic case of this problem, where edge weights are subject to change and the geometric containers have to be updated, and show that our new methods are 2 to 3 times faster than recomputing everything from scratch. Finally, we discuss the software framework that we developed to realize the implementations of all of our variants of Dijkstra’s algorithm. Such a framework is not trivial to achieve as our goal was to maintain a common code base that is at the same time small, efficient, and flexible, as we wanted to enhance and combine several variants in any possible way.