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46
Collision Detection Between Geometric Models: A Survey
 In Proc. of IMA Conference on Mathematics of Surfaces
, 1998
"... In this paper, we survey the state of the art in collision detection between general geometric models. The set of models include polygonal objects, spline or algebraic surfaces, CSG models, and deformable bodies. We present a number of techniques and systems available for contact determination. We a ..."
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Cited by 184 (15 self)
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In this paper, we survey the state of the art in collision detection between general geometric models. The set of models include polygonal objects, spline or algebraic surfaces, CSG models, and deformable bodies. We present a number of techniques and systems available for contact determination. We also describe several Nbody algorithms to reduce the number of pairwise intersection tests. 1 Introduction The goal of collision detection (also known as interference detection or contact determination) is to automatically report a geometric contact when it is about to occur or has actually occurred. The geometric models may be polygonal objects, splines, or algebraic surfaces. The problem is encountered in computeraided design and machining (CAD/CAM), robotics and automation, manufacturing, computer graphics, animation and computer simulated environments. Collision detection enables simulationbased design, tolerance verification, engineering analysis, assembly and disassembly, motion pla...
The Singular Value Decomposition for Polynomial Systems
, 1995
"... This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and ..."
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Cited by 82 (9 self)
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This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard's uresultant algorithm for the solution of overdetermined systems of polynomial equations to the inexactcoefficient case. We also briefly discuss an application of the modied Lazard's method to the location of singular points on approximately known projections of algebraic curves.
Computing Rectifying Homographies for Stereo Vision
, 1999
"... Image rectification is the process of applying a pair of 2 dimensional projective transforms, or homographies, to a pair of images whose epipolar geometry is known so that epipolar lines in the original images map to horizontally aligned lines in the transformed images. We propose a novel technique ..."
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Cited by 71 (0 self)
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Image rectification is the process of applying a pair of 2 dimensional projective transforms, or homographies, to a pair of images whose epipolar geometry is known so that epipolar lines in the original images map to horizontally aligned lines in the transformed images. We propose a novel technique for image rectification based on geometrically well defined criteria such that image distortion due to rectification is minimized. This is achieved by decomposing each homography into a specialized projective transform, a similarity transform, followed by a shearing transform. The effect of image distortion at each stage is carefully considered.
Matrices in Elimination Theory
, 1997
"... The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in ..."
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Cited by 45 (17 self)
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The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...
Polynomial Roots from Companion Matrix Eigenvalues
 Math. Comp
, 1995
"... In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a fir ..."
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Cited by 45 (1 self)
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In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a first order error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backwards normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backwards error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backwards error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, wher...
Solving systems of polynomial equations
 Computer Graphics and Applications 14, IEEE
, 1994
"... Current geometric and solid modeling systems use semialgebraic sets for de ning the boundaries of solid objects, curves and surfaces, geometric constraints with mating relationship in a mechanical assembly, physical contacts between objects, collision detection. It turns out that performing many of ..."
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Cited by 40 (8 self)
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Current geometric and solid modeling systems use semialgebraic sets for de ning the boundaries of solid objects, curves and surfaces, geometric constraints with mating relationship in a mechanical assembly, physical contacts between objects, collision detection. It turns out that performing many of the geometric operations on the solid boundaries or interacting with geometric constraints is reduced to nding common solutions of the polynomial equations. Current algorithms in the literature based on symbolic, numeric and geometric methods su er from robustness, accuracy or e ciency problems or are limited to a class of problems only. In this paper we present algorithms based on multipolynomial resultants and matrix computations for solving polynomial systems. These algorithms are based on the linear algebra formulation of resultants of equations and in many cases there is an elegant relationship between the matrix structures and the geometric formulation. The resulting algorithm involves singular value decompositions, eigendecompositions, Gauss elimination etc. In the context of oating point computation their numerical accuracy is well understood. We also present techniques to make use of the structure of the matrices to improve
A reordered Schur factorization method for zerodimensional polynomial systems with multiple roots
 In Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation
, 1997
"... We discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations. The principal contribution of the paper is to show how to reduce the multivariate problem to a univariate p ..."
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Cited by 38 (3 self)
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We discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations. The principal contribution of the paper is to show how to reduce the multivariate problem to a univariate problem, even in the case of multiple roots, in a numerically stable way. 1 Introduction The technique of solving systems of multivariate polynomial equations via eigenproblems has become a topic of active research (with applications in computeraided design and control theory, for example) at least since the papers [2, 6, 9]. One may approach the problem via various resultant formulations or by Grobner bases. As more understanding is gained, it is becoming clearer that eigenvalue problems are the "weakly nonlinear nucleus to which the original, strongly nonlinear task may be reduced"[13]. Early works concentrated on the case of simple roots. An example of such was the paper [5], which use...
Rapid and Accurate Contact Determination between Spline Models using ShellTrees
, 1998
"... In this paper, we present an efficient algorithm for contact determination between spline models. We make use of a new hierarchy, called ShellTree, that comprises of spherical shells and oriented bounding boxes. Each spherical shell corresponds to a portion of the volume between two concentric spher ..."
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Cited by 24 (5 self)
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In this paper, we present an efficient algorithm for contact determination between spline models. We make use of a new hierarchy, called ShellTree, that comprises of spherical shells and oriented bounding boxes. Each spherical shell corresponds to a portion of the volume between two concentric spheres. Given large spline models, our algorithm decomposes each surface into Bezier patches as part of preprocessing. At runtime it dynamically computes a tight fitting axisaligned bounding box across each Bezier patch and efficiently checks all such boxes for overlap. Using offline and online techniques for tree construction, our algorithm computes ShellTrees for Bezier patches and performs fast overlap tests between them to detect collisions. The overall approach can trade off runtime performance for reduced memory requirements. We have implemented the algorithm and tested itonlarge models, each composed of hundred ofpatches. Its performance varies with the configurations of the objects. For many complex models composed of hundreds of patches, it can accurately compute the contacts in a few milliseconds.