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56
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 90 (20 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 80 (14 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Sparse Elimination and Applications in Kinematics
 UC BERKELEY
, 1994
"... This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear mul ..."
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Cited by 52 (11 self)
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This thesis proposes efficient algorithmic solutions to problems in computational algebra and computational algebraic geometry. Moreover, it considers their application to different areas where algebraic systems describe kinematic and geometric constraints. Given an arbitrary system of nonlinear multivariate polynomial equations, its resultant serves in eliminating variables and reduces root finding to a linear eigenproblem. Our contribution is to describe the first efficient and general algorithms for computing the sparse resultant. The sparse resultant generalizes the classical homogeneous resultant and exploits the structure of the given polynomials. Its size depends only on the geometry of the input Newton polytopes. The first algorithm uses a subdivision of the Minkowski sum and produces matrix...
A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling
, 1997
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Approximate Boolean Operations on Freeform Solids
, 2001
"... In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces. ..."
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Cited by 41 (5 self)
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In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces.
Checking Geometric Programs or Verification of Geometric Structures
 IN PROC. 12TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1996
"... A program checker verifies that a particular program execution is correct. We give simple and efficient program checkers for some basic geometric tasks. We report about our experiences with program checking in the context of the LEDA system. We discuss program checking for data structures that ha ..."
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Cited by 34 (7 self)
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A program checker verifies that a particular program execution is correct. We give simple and efficient program checkers for some basic geometric tasks. We report about our experiences with program checking in the context of the LEDA system. We discuss program checking for data structures that have to rely on userprovided functions.
Controlled Perturbation for Arrangements of Polyhedral Surfaces with Application to Swept Volumes
 IN PROC. 15TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1999
"... We describe a perturbation scheme to overcome degeneracies and precision problems for algorithms that manipulate polyhedral surfaces using oating point arithmetic. The perturbation algorithm is simple, easy to program and completely removes all degeneracies. We describe a software package that imple ..."
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Cited by 31 (0 self)
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We describe a perturbation scheme to overcome degeneracies and precision problems for algorithms that manipulate polyhedral surfaces using oating point arithmetic. The perturbation algorithm is simple, easy to program and completely removes all degeneracies. We describe a software package that implements it, and report experimental results. The perturbation requires O(n log 3 n+nDK 2 ) expected time and O(n log n+nK 2 ) working storage, and has O(n) output size, where n is the total number of facets in the surfaces, K is a small constant in the input instances that we have examined, D is a constant greater than K but still small in most inputs, and both might be as large as n in `pathological' inputs. A tradeoff exists between the magnitude of the perturbation and the efficiency of the computation. Our perturbation package can be used by any application that manipulates polyhedral surfaces and needs robust input, such as solid modeling, manufacturing and robotics. We describe an app...
Robust Plane Sweep for Intersecting Segments
, 1997
"... In this paper, we reexamine in the framework of robust computation the BentleyOttmann 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 d ..."
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Cited by 30 (3 self)
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In this paper, we reexamine in the framework of robust computation the BentleyOttmann 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 exactcomputation paradigm we introduce two models of computation aimed at replacing the conventional model of realnumber 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...
A Complete Implementation for Computing General Dimensional Convex Hulls
 INT. J. COMPUT. GEOM. APPL
, 1995
"... We study two important, and often complementary, issues in the implementation of geometric algorithms, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic ..."
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Cited by 28 (9 self)
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We study two important, and often complementary, issues in the implementation of geometric algorithms, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with randomization in order to speed up the lifting of residues to an integer. We derive general error bounds and discuss the implementation of this approach in our generaldimension convex hull program. The use of perturbations as a method to cope with input degeneracy is also illustrated. We present the implementation of a computationally efficient scheme that, moreover, greatly simplifies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Starting in the context of a specific application in robotics, we examine the complexity of p...
Efficient perturbations for handling geometric degeneracies
 ALGORITHMICA
, 1997
"... This article defines input perturbations so that an algorithm designed under certain restrictions on the input can execute on arbitrary instances. A syntactic definition of perturbations is proposed and certain properties are specified under which an algorithm executed on perturbed input produces an ..."
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Cited by 25 (2 self)
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This article defines input perturbations so that an algorithm designed under certain restrictions on the input can execute on arbitrary instances. A syntactic definition of perturbations is proposed and certain properties are specified under which an algorithm executed on perturbed input produces an output from which the exact answer can be recovered. A general framework is adopted for linear perturbations, which are efficient from the point of view of worstcase complexity. The deterministic scheme of Emiris and Canny [1] was the first efficient scheme and is extended in a consistent manner, most notably to the InSphere primitive. We introduce a variant scheme, applicable to a restricted class of algorithms, which is almost optimal in terms of algebraic as well as bit complexity. Neither scheme requires any symbolic computation and both are simple to use as illustrated by our implementation of a convex hull algorithm in arbitrary dimension. Empirical results and a concrete application in robotics are presented.