227 research outputs found

    Rational Hausdorff Divisors: a New approach to the Approximate Parametrization of Curves

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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

    The Relation Between Offset and Conchoid Constructions

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    The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided

    First Steps Towards Radical Parametrization of Algebraic Surfaces

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    We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.Comment: 31 pages, 7 color figures. v2: added another case of genus

    Grobner Basis Computation of Drazin Inverses with Multivariate Rational Function Entries

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    In this paper we show how to apply Grobner bases to compute the Drazin inverse of a matrix with multivariate rational functions as entries. When the coeficients of the rational functions depend on parameters, we give suficient conditions for the Drazin inverse to specialize properly. In addition, we extend the method to weighted Drazin inverses. We present an empirical analysis that shows a good timing performance of the method

    Computation of Moore-Penrose generalized inverses of matrices with meromorphic function entries

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    J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)In this paper, given a field with an involutory automorphism, we introduce the notion of Moore-Penrose field by requiring that all matrices over the field have Moore-Penrose inverse. We prove that only characteristic zero fields can be Moore-Penrose, and that the field of rational functions over a Moore-Penrose field is also Moore-Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore-Penrose inverse is the Moore-Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromeorphic functions being invariant by the involutory automorphism, computes its Moore-Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.Ministerio de EconomĂ­a y CompetitividadEuropean Regional Development Fun
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