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The topology of the Voronoi diagram of three lines
 PROCEEDINGS OF SYMPOSIUM ON COMPUTATIONAL GEOMETRY, ACM PRESS, SOUTHKOREA
, 2007
"... We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four ..."
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Cited by 14 (4 self)
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We give a complete description of the Voronoi diagram, in R³, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a nonsingular quartic or of a nonsingular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique, which relies heavily upon modern tools of computer algebra, and is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semialgebraic tests for separating the two connected components of each twodimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semialgebraic tests.
The predicates for the Voronoi diagram of ellipses
 In Proc. 22th Annual ACM Symp. on Computational Geometry
, 2006
"... This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exter ..."
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Cited by 13 (8 self)
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This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to nonintersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivisionbased algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.
MACAULAY STYLE FORMULAS FOR SPARSE RESULTANTS
, 2002
"... We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials. ..."
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Cited by 12 (0 self)
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We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.
On the Topology of Planar Algebraic Curves
"... We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e cienc ..."
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Cited by 7 (1 self)
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We introduce a method to compute the topology of planar algebraic curves. The curve may not be in generic position and may have vertical asymptotes. The algebraic tools are rational univariate representation for zerodimentional ideals and multiplicities in these ideals. Experiments show the e ciency of our algorithm. 1
SYMMETRIC TENSOR DECOMPOSITION
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 6 (0 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank. 1.
Author manuscript, published in "Linear Algebra and Applications 433, 1112 (2010) 18511872" SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variab ..."
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Abstract. We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasiHankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in nongeneric cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for fl ..."
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Abstract. The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.