Computation of the topology of real algebraic space curves

Alcázar, Juan Gerardo; Sendra, J. Rafael

doi:10.1016/j.jsc.2005.01.006

Cited by 46 publications

(33 citation statements)

References 15 publications

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“…If this property is not true for a certain pair of edges from a 3-dimensional graph, then we report a failure message during runtime. [1], [12], the 3-dimensional graph computed as the piecewise linear approximation of an implicitly defined space algebraic curve is called the topology of the curve. We use the Axel [13] free algebraic geometric modeler to compute the 3-dimensional graph as presented in [8], [10].…”

Section: Data Structuresmentioning

confidence: 99%

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

Hodorog

Mourrain

Schicho

2011

Computational Science and Its Applications - ICCSA 2011

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Abstract. We report on an adapted version of the Bentley-Ottmann algorithm for computing all the intersection points among the edges of the projection of a three-dimensional graph. This graph is given as a set of vertices together with their space Euclidean coordinates, and a set of edges connecting them. More precisely, the three-dimensional graph represents the approximation of a closed and smooth implicitly defined space algebraic curve, that allows us a simplified treatment of the events encountered in the Bentley-Ottmann algorithm. As applications, we use the adapted algorithm to compute invariants for each singularity of a plane complex algebraic curve, i.e. the Alexander polynomial, the Milnor number, the delta-invariant, etc.

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Section: Data Structuresmentioning

confidence: 99%

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

Hodorog

Mourrain

Schicho

2011

Computational Science and Its Applications - ICCSA 2011

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“…If it is not, we can put it in pseudo-generic position by a change of coordinates. In fact, there is only a finite number of bad changes of coordinates of the form X := X + λZ; Y := Y + µZ; Z := Z, with λ, µ ∈ É * such that if CÊ is not in pseudo-generic position then the transformed curve remains in a non-pseudogeneric position [1].…”

Section: Propositionmentioning

confidence: 99%

“…The corresponding α is called a x-critical value. This notion of genericity also appears in a slightly more restrictive form in [1]. The aim of the next section is to give an algorithm to certify the third point of the previous definition of generic position.…”

Section: Propositionmentioning

confidence: 99%

“…The problem of computing the topology of space curves has been less investigated. In [1], Alcázar and Sendra give a symbolic-numeric algorithm for reduced space curves using subresultant and GCD computations of approximated polynomials. If their approach gives good practical results however it doesn't give a rigorous proof that a sufficient precision is selected for all inputs in the computation of GCD of approximated polynomials.…”

Section: Introductionmentioning

confidence: 99%

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On the computation of the topology of a non-reduced implicit space curve

Daouda

Mourrain

Ruatta

2008

Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation

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An algorithm is presented for the computation of the topology of a non-reduced space curve defined as the intersection of two implicit algebraic surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the original space curve.The algorithm is designed to provide the exact result for all inputs. It's a symbolic-numeric algorithm based on subresultant computation. Simple algebraic criteria are given to certify the output of the algorithm.The algorithm uses only one projection of the non-reduced space curve augmented with adjacency information around some "particular points" of the space curve.The algorithm is implemented with the Mathemagix Computer Algebra System (CAS) using the SYNAPS library as a backend.

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“…They decompose the problem into several sub-problems, and sort these problems according to the curve topology [15,16]. The decomposition terminates if suitable approximating primitives can be generated for each sub-problem [17].…”

Section: Introductionmentioning

confidence: 99%

Approximating Algebraic Space Curves by Circular Arcs

Béla

Jüttler

2012

Curves and Surfaces

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Abstract. We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve.

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Computation of the topology of real algebraic space curves

Cited by 46 publications

References 15 publications

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

An Adapted Version of the Bentley-Ottmann Algorithm for Invariants of Plane Curves Singularities

On the computation of the topology of a non-reduced implicit space curve

Approximating Algebraic Space Curves by Circular Arcs

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