. We use known data structures for ray-shooting and linear-programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f -face convex hull of an n-point set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 1-1/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 1-1/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2-# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in d-dimen...

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.

Genus-reducing simplifications are important in constructing multiresolution hierarchies for level-of-detail-based rendering, especially for datasets that have several relatively small holes, tunnels, and cavities. We present a genus-reducing simplification approach that is complementary to the existing work on genus-preserving simplifications. We propose a simplification framework in which genus-reducing and genus-preserving simplifications alternate to yield much better multiresolution hierarchies than would have been possible by using either one of them. In our approach we first identify the holes and the concavities by extending the concept of #- hulls to polygonal meshes under the L1 distance metric and then generate valid triangulations to fill them. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation --- Display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling --- Curve, surface, solid, and object represent...

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the "shape" of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three- dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case.

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This dissertation explores some techniques for automatic approximation of geometric objects. My thesis is that using and extending concepts from computational geometry can help us in devising efficient and parallelizable algorithms for automatically constructing useful detail hierarchies for geometric objects. We have demonstrated this by developing new algorithms for two kinds of geometric approximation problems that have been motivated by a single driving problem --- the efficient computation and display of smooth solvent-accessible molecular surfaces. The applications of these detail hierarchies are in biochemistry and computer graphics. The smooth solvent-accessible surface of a molecule is useful in studying the structure and interactions of proteins, in particular for attacking the protein-substrate docking problem. We have developed a parallel linear-time algorithm for computing molecular surfaces. Molecular surfaces are equivalent to the weighted ff-hulls. Thus our work is pot...

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean d-dimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...

We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that they have bounded-depth. Metrologists in this field have mounted a quest for a reference software which will impose some certainty in a confusing market of metrology software. The use of exact computation in the reference software will solve many intractable difficulties associated with current approaches. In short, here is a practical area in which CG and exact computation can have a real impact. 1 Introduction Researchers in Computational Geometry (CG) have always been convinced that their subject is relevant to a variety of application areas. But CG'ers have often assumed that the application areas would come to CG to find answers to their questions. To what extent is this valid? I will d...

We present a topology simplifying approach that can be used for genus reductions, removal of protuberances, and repair of cracks in polygonal models in a unified framework. Our work is complementary to the existing work on geometry simplification of polygonal datasets and we demonstrate that using topology and geometry simplifications together yields superior multiresolution hierarchies than is possible by using either of them alone. Our approach can also address the important issue of repair of cracks in polygonal models as well as for rapid identification and removal of protuberances based on internal accessibility in polygonal models. Our approach is based on identifying holes and cracks by extending the concept of #-shapes to polygonal meshes under the L1 distance metric. We then generate valid triangulations to fill them using the intuitive notion of sweeping a L1 cube over the identified regions. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture...