Topology and arrangement computation of semi-algebraic planar curves

Alberti, Lionel; Mourrain, Bernard; Wintz, Julien

doi:10.1016/j.cagd.2008.06.009

Cited by 47 publications

(50 citation statements)

References 34 publications

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“…For singular points, we solve the system of singular points with additional constraints. Note that the overall method to compute the topology at critical points is not new, it is described in full details in [49], see also for example [1,4,10,33,40,41] for closely related approaches for curves in non-generic position. The novelty appears in the way we compute multiplicities in this context; once again we avoid computing sub-resultant sequences.…”

Section: Our Contributionsmentioning

confidence: 99%

“…, x n ] which is injective on V (I ); γ is called the separating polynomial of the RUR. 4 Note that a random degree-one polynomial in x 1 , . .…”

Section: Rational Univariate Representationmentioning

confidence: 99%

“…Note finally that another approach that avoids expensive algebraic computations is to compute the critical or singular points using subdivision methods [4,12]. The major drawback of these methods is that, in order to certify the results, the subdivision has, in general, to reach a separation bound (certification can also be achieved by solving algebraic systems by other means).…”

mentioning

confidence: 99%

“…To our knowledge, almost all methods for computing the topology of a curve compute the critical points of the curve and associate corresponding vertices in the graph. (Refer to [4,12] for recent subdivision methods that avoid the computation of non-singular critical points.) However, it should be stressed that almost all methods do not necessarily compute the critical points for the specified x-direction.…”

mentioning

confidence: 99%

“…Previous Work There have been many papers addressing the problem of computing the topology of algebraic plane curves (or closely related problems) defined by a bivariate polynomial with rational coefficients [1,3,4,8,10,18,19,22,26,30,32,33,40,41,46,48]. Most of the algorithms assume generic position for the input curve.…”

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confidence: 99%

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On the Topology of Real Algebraic Plane Curves

et al. 2010

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We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves. Fig. 1 a Curve of equation 16x 5 − 20x 3 + 5x − 4y 3 + 3y = 0 plotted in maple, b its isotopic graph computed by isotop, and c their overlay

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Section: Our Contributionsmentioning

confidence: 99%

“…, x n ] which is injective on V (I ); γ is called the separating polynomial of the RUR. 4 Note that a random degree-one polynomial in x 1 , . .…”

Section: Rational Univariate Representationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the Topology of Real Algebraic Plane Curves

et al. 2010

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On the Topology of the Intersection Curve of Two Real Parameterized Algebraic Surfaces

Diatta

Ruatta

2022

Trends in Mathematics

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Singular Zeros of Polynomial Systems

Mantzaflaris

Mourrain

2014

SAGA – Advances in ShApes, Geometry, and Algebra

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Singular zeros of systems of polynomial equations constitute a bottleneck when it comes to computing, since several methods relying on the regularity of the Jacobian matrix of the system do not apply when the latter has a non-trivial kernel. Therefore they require special treatment. The algebraic information regarding an isolated singularity can be captured by a finite, local basis of differentials expressing the multiplicity structure of the point.In the present article, we review some available algebraic techniques for extracting this information from a polynomial ideal. The algorithms for extracting the, so called, dual basis of the singularity are based on matrix-kernel computations, which can be carried out numerically, starting from an approximation of the zero in question.The next step after obtaining the multiplicity structure is to deflate the root, that is, construct a new system in which the singularity is eliminated. Having a deflated system allows to refine the solution fast and to high accuracy, since the Jacobian matrix is regular and all the usual machinery, e.g. Newton's method or existence and unicity criteria may be applied. Standard verification methods, based e.g. on interval arithmetic and a fixed point theorem, can then be employed to certify that there exists a unique perturbed system with a singular root in the domain.

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Topology and arrangement computation of semi-algebraic planar curves

Cited by 47 publications

References 34 publications

On the Topology of Real Algebraic Plane Curves

On the Topology of Real Algebraic Plane Curves

On the Topology of the Intersection Curve of Two Real Parameterized Algebraic Surfaces

Singular Zeros of Polynomial Systems

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