Browsing by Subject "Subresultants"

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    Avoiding the general position condition whencomputing the topology of a real algebraic planecurve defined implicitly
    (Springer Nature, 2023) Caravantes, Jorge; Gema M., Diaz–Toca; Gonzalez–Vega, Laureano; Ingeniería y Tecnología de Computadores
    The problem of computing the topology of curves has re-ceived special attention from both Computer Aided Geometric Designand Symbolic Computation. It is well known that the general positioncondition simplifies the computation of the topology of a real algebraicplane curve defined implicitly since, under this assumption, singularpoints can be presented in a very convenient way for that purpose. Herewe will show how the topology of cubic, quartic and quintic plane curvescan be computed in the same manner even if the curve is not in gen-eral position, avoiding thus coordinate changes. This will be possibleby applying new formulae, derived from subresultants, which describemultiple roots of univariate polynomials as rational functions of the con-sidered polynomial coefficients. We will also characterize those higherdegree curves where this approach can be used and use this technique todescribe the curve arising when intersecting two ellipsoids.
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    Solving the interference problem for ellipses and ellipsoids: New formulae.
    (Elsevier. Journal of Computational and Applied Mathematics 407 (2022) ., 2022-06) Caravantes, Jorge; Díaz Toca, Gema M.; Fioravanti, Mario; González Vega, Laureano; Ingeniería y Tecnología de Computadores
    The problem of detecting when two moving ellipses or ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc., where ellipses and ellipsoids are often used for modelling (and/or enclosing) the shape of the objects under consideration. By analysing symbolically the sign of the real roots of the characteristic polynomial of the pencil defined by two ellipses/ellipsoids A and B given by XTAX = 0 and XTBX = 0, we derive new formulae characterising when A and B overlap, are separate, or touch each other externally. This characterisation is defined by a minimal set of polynomial inequalities depending only on the entries of A and B so that we need only compute the characteristic polynomial of the pencil defined by A and B, det(TA + B), and not the intersection points between them. Compared with the best available approach dealing with this problem, the new formulae involve a smaller set of polynomials and less sign conditions. As an application, this characterisation provides also a new approach for exact collision detection of two moving ellipses or ellipsoids since the analysis of the univariate polynomials (depending on the time) in the previously mentioned formulae provides the collision events between them.