Unidad docente MatemáticasMATEMATIChttp://hdl.handle.net/10017/2812023-11-29T12:07:02Z2023-11-29T12:07:02ZSymmetries and similarities of planar algebraic curves using harmonic polynomialsAlcázar Arribas, Juan GerardoLávička, MiroslavVršek, Janhttp://hdl.handle.net/10017/585742023-11-29T01:18:02Z2019-09-01T00:00:00ZSymmetries and similarities of planar algebraic curves using harmonic polynomials
Alcázar Arribas, Juan Gerardo; Lávička, Miroslav; Vršek, Jan
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a similarity transformation. Both algorithms are based on the fact, well-known in Harmonic Analysis, that the Laplacian commutes with orthogonal transformations, and on efficient algorithms to find the symmetries / similarities of a harmonic algebraic curve / two given harmonic algebraic curves. In fact, we show that, except for some special cases, the problem can be reduced to the harmonic case.
2019-09-01T00:00:00ZAffine equivalences, isometries and symmetries of ruled rational surfacesAlcázar Arribas, Juan GerardoQuintero, Emilyhttp://hdl.handle.net/10017/585702023-11-29T01:18:02Z2020-01-15T00:00:00ZAffine equivalences, isometries and symmetries of ruled rational surfaces
Alcázar Arribas, Juan Gerardo; Quintero, Emily
An algorithmic method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations. The algorithm works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method translates the problem into parameter space, and relies on polynomial system solving. Geometrically, the problem is related to finding the projective equivalences between two projective curves (corresponding to the directions of the rulings of the surfaces). This problem was recently addressed in a paper by Hauer and Jüttler, and we exploit the ideas by these authors in the algorithm presented in this paper. The general idea for affine equivalences is adapted to computing the isometries between two rational ruled surfaces, and the symmetries of a given rational ruled surface. The efficiency of the method is shown through several examples.
2020-01-15T00:00:00ZA new method to detect projective equivalences and symmetries of rational 3D curvesGözütok, UğurÇoban, Hüsnü AnılSağıroğlu, YaseminAlcázar Arribas, Juan Gerardohttp://hdl.handle.net/10017/585672023-11-29T01:18:02Z2023-02-01T00:00:00ZA new method to detect projective equivalences and symmetries of rational 3D curves
Gözütok, Uğur; Çoban, Hüsnü Anıl; Sağıroğlu, Yasemin; Alcázar Arribas, Juan Gerardo
We present a new approach to detect projective equivalences and symmetries between two rational parametric curves properly parametrized. In order to do this, we introduce two rational functions that behave nicely for Möbius transformations, which are the transformations in the parameter space associated with the projective equivalences between the curves. The Möbius transformations are found by first computing the gcd of two polynomials built from these two functions, and then searching for a special type of factors, ?Möbius-like?, of this gcd. The projective equivalences themselves are easily computed from the Möbius transformations. In particular, and unlike previous approaches, we avoid solving big polynomial systems. The algorithm has been implemented in Maple? (2021), and evidences of its efficiency as well as a comparison with previous approaches are given.
2023-02-01T00:00:00ZAffine equivalences of trigonometric curvesAlcázar Arribas, Juan GerardoQuintero, Emilyhttp://hdl.handle.net/10017/585532023-11-28T01:14:51Z2020-08-24T00:00:00ZAffine equivalences of trigonometric curves
Alcázar Arribas, Juan Gerardo; Quintero, Emily
We provide an efficient algorithm to detect whether two given trigonometric curves, i.e. two parametrized curves whose components are truncated Fourier series, in any dimension, are affinely equivalent, i.e. whether there exists an affine mapping transforming one of the curves onto the other. If the coefficients of the parametrizations are known exactly (the exact case), the algorithm boils down to univariate gcd computation, so it is efficient and fast. If the coefficients of the parametrizations are known with finite precision, e.g. floating point numbers (the approximate case), the univariate gcd computation is replaced by the computation of singular values of an appropriate matrix. Our experiments show that the method works well, even for high degrees.
2020-08-24T00:00:00Z