Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...

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We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2- and 3-dimensional roundness primitives, deriving results that improve previous approaches in several respects, including problem definition, running time, underlying computational model, and dimensionality of the input. We also study methods for determining the width of a d-dimensional point set, which corresponds to the metrology notion of "flatness," giving an approximation method that can serve as a fast exactcomputation filter for this metrology primitive. Finally, we report on experimental results derived from implementation and testing, particularly in 3-space, of our approximation algorithms, including several heuristics designed to significantly speed-up the computations in practice.

. We describe the design and discussthe performance of a parallel elastic wave propagation simulator that is being used to model earthquake-induced ground motion in large sedimentary basins. The components of the system include mesh generators, a mesh partitioner, a parceler, and a parallel code generator, as well as parallel numerical methods for applying seismic forces, incorporating absorbing boundaries, and solving the discretized wave propagation problem. We discuss performance on the Cray T3D for unstructured mesh wave propagation problems of up to 77 million tetrahedra. By paying careful attention to each step of the process, we obtain excellent performance despite the highly irregular structure of the problem. The mesh generator, partitioner, parceler, and code generator collectively form an integrated toolset called Archimedes, which automates the solution of unstructured mesh PDE problems on parallel computers, and is being used for other unstructured mesh applications beyond...

In this paper, we reexamine in the framework of robust computation the Bentley-Ottmann algorithm for reporting intersecting pairs of segments in the plane. This algorithm has been reported as being very sensitive to numerical errors. Indeed, a simple analysis reveals that it involves predicates of degree 5, presumably never evaluated exactly in most implementation. Within the exact-computation paradigm we introduce two models of computation aimed at replacing the conventional model of real-number arithmetic. The first model (predicate arithmetic) assumes the exact evaluation of the signs of algebraic expressions of some degree, and the second model (exact arithmetic) assumes the exact computation of the value of...

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. 1

Let P be a polyhedral subdivision in R 3 with a total of n faces. We show that there is an embedding oe of the vertices, edges, and facets of P into a subdivision Q, where every vertex coordinate of Q is an integral multiple of 2 \Gammadlog 2 n+2e . For each face f of P , the Hausdorff distance in the L1 metric between f and oe(f) is at most 3/2. The embedding oe preserves or collapses vertical order on faces of P . The subdivision Q has O(n 4 ) vertices in the worst case, and can be computed in the same time. 1 Introduction Geometric algorithms are usually described in the "real-number RAM" model of computation, where arithmetic operations on real numbers have unit cost. A programmer implementing a geometric algorithm must find some substitution for real arithmetic. The substitution of exact arithmetic on a subset of the reals, say the integers or the rationals, avoids the difficulties that can arise from naive substitution of floating-point arithmetic [4, 12, 14, 15]. The subs...

Computational geometry has produced an impressive wealth of efficient algorithms.

We describe a distributed memory parallel Delaunay refinement algorithm for simple polyhedral domains whose constituent bounding edges and surfaces are separated by angles between 90 o to 270 o inclusive. With these constraints, our algorithm can generate meshes containing tetrahedra with circumradius to shortest edge ratio less than 2, and can tolerate more than 80 % of the communication latency caused by unpredictable and variable remote gather operations. Our experiments show that the algorithm is efficient in practice, even for certain domains whose boundaries do not conform to the theoretical limits imposed by the algorithm. The algorithm we describe is the first step in the development of much more sophisticated guaranteed–quality parallel mesh generation algorithms.

We consider whether restricted sets of geometric predicates support efficient algorithms to solve line and curve segment intersection problems in the plane. Our restrictions are based on the notion of algebraic degree, proposed by Preparata and others as a way to guide the search for efficient algorithms that can be implemented in more realistic computational models than the Real RAM.