Topology of real algebraic space curves

Kahoui, M’hammed El

doi:10.1016/j.jsc.2007.10.008

Cited by 31 publications

(30 citation statements)

References 14 publications

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“…This question is interesting in its own right, but it also plays an important role in many higher-level algorithms, such as computing the topology of plane and space curves [13,8] or solving general polynomial systems [18].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the complexity of solving bivariate systems

Lebreton

Mehrabi

Schost

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the complexity of solving bivariate systems

Lebreton

Mehrabi

Schost

2013

Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

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“…This notion of genericity also appears in [12] and in a slightly more restrictive form in [6]. Previous approaches succeed if genericity conditions are satisfied, but they do not guarantee to reject the curve if they are not; i.e, it does not decide genericity.…”

Section: Topology Of a Plane Algebraic Curvementioning

confidence: 99%

“…A wide literature exists on the computation of the topology of plane curves ( [8], [7], [10], [11], [12], [6] Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.…”

Section: Introductionmentioning

confidence: 99%

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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“…The algorithm we propose is inspired by El Kahoui's algorithm for the topology of a space curve [4]; we also use ideas from [7,3], that allow us to replace computations with real algebraic numbers by manipulations on isolating boxes.…”

mentioning

confidence: 99%

“…Using results from [4], we deduce the topology of the space curve from the one computed in Step 1, and use it to answer connectivity queries on the space curve.…”

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confidence: 99%

Connectivity queries on curves in Rn

Islam

Poteaux

2011

ACM Commun. Comput. Algebra

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Canny introduced the notion of roadmap in [2], as a way to study connectivity properties of semialgebraic sets (which appear for instance in motion planning problems).A roadmap R of a semi-algebraic set V is a curve contained in V , that has a non-empty and connected intersection with each connected component of S. Given two query points A, B on V , it is possible to construct a roadmap that contains both of them. Then, A and B belong to the same connected component of V if and only if they are on the same connected component of R. Thus, roadmaps allow one to reduce connectivity queries on semi-algebraic sets to connectivity queries on curves.Let n be the dimension of the ambient space, and let X 1 , . . . , X n be coordinates in C n . Following Canny's algorithm, and improvements by Basu, Pollack and Roy [1], the roadmap algorithm from [5] computes the following:1. two linear forms η = η 1 X 1 + · · · + η n X n and ϑ = ϑ 1 X 1 + · · · + ϑ n X n , with coefficients in Q 2. polynomials q, q 0 , . . . , q n in Q[T, U ] where T and U are indeterminates.Let Z ⊂ C n be the constructible set defined byThen, the roadmap R is obtained as C ∩ R n , where C ⊂ C n is the algebraic curve obtained as the Zariski closure of Z.In this work, we consider this roadmap as our input. Given (q, q 0 , . . . , q n ) and η, ϑ, as well as two query points A, B on R, our question is to decide whether A and B are on the same connected component of R. To our knowledge, no previous work directly addresses this question. A close reference is in [6], which however considers a more general input (given by means of a regular chain), and relies on Puiseux series computations.The algorithm we propose is inspired by El Kahoui's algorithm for the topology of a space curve [4]; we also use ideas from [7,3], that allow us to replace computations with real algebraic numbers by manipulations on isolating boxes.Our algorithm, as well as in El Kahoui's, requires that the input curve be in general position. The genericity requirements are of a geometric nature (e.g., there should be no point on R with a tangent orthogonal to the η, ϑ-plane, etc).Of course, these conditions can be ensured by means of a generic enough change of coordinates A; we can also suppose that the linear forms η, ϑ are X 1 , X 2 . We give a precise cost estimate for the application of this change of coordinates; we also prove that the set of all unlucky A is contained in a strict algebraic subset of GL n of degree δ O(1) , where δ is the degree of C. Using Zippel-Schwartz's lemma, this allows us to determine the probability of success of finding a generic enough change of coordinates A in a large finite subset of GL n .Supposing that the chosen change of coordinates is generic, our algorithm works in three steps:117

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Topology of real algebraic space curves

Cited by 31 publications

References 14 publications

On the complexity of solving bivariate systems

On the complexity of solving bivariate systems

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

Connectivity queries on curves in Rn

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