Many algorithms developed in computational geometry are needlessly complicated and slow because they have to be prepared for very complicated, hypothetical inputs. To avoid this, realistic models are needed that describe the properties that realistic inputs have, so that algorithms can de designed that take advantage of these properties. This can lead to algorithms that are provably efficient in realistic situations. We obtain some fundamental results in this research direction. In particular, we have the following results. ffl We show the relations between various models that have been proposed in the literature. ffl For several of these models, we give algorithms to compute the model parameter(s) for a given scene; these algorithms can be used to verify whether a model is appropriate for typical scenes in some application area. ffl As a case study, we give some experimental results on the appropriateness of some of the models for one particular type of scenes often encountered in ...

We present a general technique for approximating various descriptors of the extent of a set P of n points in R . For a given extent measure and a parameter " > 0, it computes in time O(n + 1=" ) a subset Q P of size 1=" , with the property that (1 ")(P ) (Q) (P ). The speci c applications of our technique include "-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of P , (ii) maintaining all the previous measures for a set of moving points, and (iii) tting spheres and cylinders through a point set P . Our algorithms are considerably simpler, and faster in many cases, than the known algorithms.

Ray (segment) shooting is the problem of determining the first intersection between a ray (directed line segment) and a collection of polygonal or polyhedral obstacles. In order to process queries efficiently, the set of obstacle polyhedra is usually preprocessed into a data structure. In this paper, we propose a query-sensitive data structure for ray shooting, which means that the performance of our data structure depends on the "local" geometry of obstacles near the query segment. We measure the complexity of the local geometry near the segment by a parameter called the simple cover complexity , denoted by scc(s) for a segment s. Our data structure consists of a subdivision that partitions the space into a collection of polyhedral cells of O(1) complexity. We answer a segment shooting query by walking along the segment through the subdivision. Our first result is that, for any fixed dimension d, there exists a simple hierarchical subdivision in which no query segment s int...

Abstract. The paradigm of coresets has recently emerged as a powerful tool for efficiently approximating various extent measures of a point set P. Using this paradigm, one quickly computes a small subset Q of P, called a coreset, that approximates the original set P and and then solves the problem on Q using a relatively inefficient algorithm. The solution for Q is then translated to an approximate solution to the original point set P. This paper describes the ways in which this paradigm has been successfully applied to various optimization and extent measure problems. 1.

In this paper we consider the following problem: given two general polyhedra of complexity n, one of which is moving translationally or rotating about a fixed axis, determine the first collision (if any) between them. We present an algorithm with running time O(n 8=5+ffl ) for the case of translational movements and running time O(n 5=3+ffl ) for rotational movements, where ffl is an arbitrary positive constant. This is the first known algorithm with sub-quadratic running time. Universitat des Saarlandes, Fachbereich 14, Informatik, Im Stadtwald, D-66041 Saarbrucken, Germany. E-mail: schoemer@cs.uni-sb.de. y Max-Planck-Institut fur Informatik, Im Stadtwald, D-66123 Saarbrucken, Germany. E-mail: thiel@mpi-sb.mpg.de. This author was supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). 1 1 Introduction The demands on quality, security and higher production capacity in manufacturing increase the need for automation during the phase...

In geometric range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

Let F be a collection of n bivariate algebraic functions of constant maximum degree. We show that the combinatorial complexity of the vertical decomposition of the k-level of the arrangement A(F) is O(k 3+" /(n=k)), for any " ? 0, where /(r) is the maximum complexity of the lower envelope of a subset of at most r functions of F . This bound is nearly optimal in the worst case, and implies the existence of shallow cuttings, in the sense of [51], of small size in arrangements of bivariate algebraic functions. We also present numerous applications of these results, including: (i) data structures for several generalized three-dimensional range searching problems; (ii) dynamic data structures for planar nearest and farthest neighbor searching under various fairly general distance functions; (iii) an improved (near-quadratic) algorithm for minimum-weight bipartite Euclidean matching in the plane; and (iv) efficient algorithms for certain geometric optimization problems in static...

We propose a fast algorithm for rendering general irregular grids. Our method uses a sweep-plane approach to accelerate ray casting, and can handle disconnected and nonconvex (even with holes) unstructured irregular grids with a rendering cost that decreases as the “disconnectedness” decreases. The algorithm is carefully tailored to exploit spatial coherence even if the image resolution differs substantially from the object space resolution. In this paper, we establish the practicality of our method through experimental results based on our implementation, and we also provide theoretical results, both lower and upper bounds, on the complexity of ray casting of irregular grids.

Abstract. Let F be a collection of nd-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ε>0. For d = 3, by combining this algorithm with the point-location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n 3+ε), for any ε>0, a data structure of size O(n 3+ε) that, for any query point q, can determine in O(log 2 n) time the function(s) of F that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the “biggest stick ” in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in randomized expected time O(n 17/11+ε), for any ε>0, improving previous solutions that run in time O(n 8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n 3+ε) storage and preprocessing time, for any ε>0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

A visibility ordering of a set of objects, from a given viewpoint, is a total order on the objects such that if object a obstructs object b,thenb precedes a in the ordering. Such orderings are extremely useful for rendering volumetric data. We present an algorithm that generates a visibility ordering of the cells of an unstructured mesh, provided that the cells are convex polyhedra and nonintersecting, and that the visibility ordering graph does not contain cycles. The overall mesh may be nonconvex and it may have disconnected components. Our technique employs the sweep paradigm to determine an ordering between pairs of exterior (mesh boundary) cells which can obstruct one another. It then builds on Williams' MPVO algorithm [33] which exploits the ordering implied by adjacencies within the mesh. The partial ordering of the exterior cells found by sweeping is used to augment the DAG created in Phase II of the MPVO algorithm. Our method thus removes the assumption of the MPVO algorithm t...