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

Hodorog, Mădălina; Mourrain, Bernard; Schicho, Josef

doi:10.1007/978-3-642-21931-3_10

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…We will not be interested in the type of the singularity; methods exist that can identify the type of a given singularity, eg. the symbolic-numeric method for Puiseux expansions of [88] or the works based on genus computation [53,54,55].…”

Section: Geometry Around a Singularitymentioning

confidence: 99%

Robust algebraic methods for geometric computing

Mantzaflaris

2012

ACM Commun. Comput. Algebra

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Geometric computation in computer aided geometric design and solid modeling calls for solving non-linear polynomial systems in an approximate-yet-certified manner. We introduce new subdivision algorithms that tackle this fundamental problem. In particular, we generalize the univariate so-called continued fraction solver to general dimension. Fast bounding functions, unicity tests, projection and preconditioning are employed to speed up convergence. Apart from practical experiments, we provide theoretical bit complexity estimates, as well as bounds in the real RAM model, by means of real condition numbers.A main bottleneck for any real solving method is singular isolated points. We employ local inverse systems and certified numerical computations to provide certification criteria to treat singular solutions. In doing so, we are able to check existence and uniqueness of singularities of a given multiplicity structure using verification methods, based on interval arithmetic and fixed point theorems.Two major geometric applications are undertaken. First, the approximation of planar semialgebraic sets, commonly occurring in constraint geometric solving. We present an efficient algorithm to identify connected components and, for a given precision, to compute polygonal and isotopic approximation of the exact set.Second, we present an algebraic framework to compute generalized Voronoi diagrams, that is applicable to any diagram type in which the distance from a site can be expressed by a bi-variate polynomial function (anisotropic, power diagram etc). In cases where this is not possible (eg. Apollonius diagram, VD of ellipses and so on), we extend the theory to implicitly given distance functions. 237

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Section: Geometry Around a Singularitymentioning

confidence: 99%

Robust algebraic methods for geometric computing

Mantzaflaris

2012

ACM Commun. Comput. Algebra

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“…We call approximate to an algorithm solving a problem of the above type; a solution for the illustrating example on polynomial factorization is given in [7]. Some papers treating this type of problems with the same, or similar, strategy are [3], [4] [5], [8], [9], [10], [11], [13], [14], [15], [17], [20]; see also [16].…”

Section: Introductionmentioning

confidence: 99%

Rational Hausdorff divisors: A new approach to the approximate parametrization of curves

Rueda

Sendra

2014

Journal of Computational and Applied Mathematics

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In this paper we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions to the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.

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Computing the equisingularity type of a pseudo-irreducible polynomial

Poteaux

Weimann

2020

AAECC

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

Cited by 4 publications

References 9 publications

Robust algebraic methods for geometric computing

Robust algebraic methods for geometric computing

Rational Hausdorff divisors: A new approach to the approximate parametrization of curves

Computing the equisingularity type of a pseudo-irreducible polynomial

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