Browsing by Subject "Planar rational curves"

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    Computing the topology of the image of a parametric planar curve under a birational transformation
    (2023) Díaz Toca, Gema M.; Gerardo Alcázar, Juan; Ingeniería y Tecnología de Computadores
    We provide a method to compute the topology of the image of a parametric curve under a birational mapping of the plane. The method proceeds by exploiting as much as possible the initial parametrization in order to reduce the computational cost. The self intersections of the image curve are derived from points in the image where the inverse of the birational mapping is not defined. In order to compute these points, we prove a result characterizing birational planar mappings, together with an algorithm to compute the inverse of a birational mapping. We apply the method when the original curve is rational, in which case the image of the curve is also rational but with a higher degree, and when the original curve is an exp-log-arctan function. In this last case the image is a non-rational curve admitting an analytic parametrization, a problem not treated in the literature so far
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    On the square-freeness of the offset equation to a rational planar curve
    (World Scientific, 2018-04-04) Díaz Toca, Gema M.; Alcázar, Juan Gerardo; Caravantes, Jorge; Ingeniería y Tecnología de Computadores
    It is well known that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the non-extraneous component of this resultant has at most two irreducible factors. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be com- puted in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the off- set has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the non-extraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.