Topology of 2D and 3D Rational Curves

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.Comment: 26 pages, 19 figure

On the shape of curves that are rational in polar coordinates

In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates, i.e. which are defined by means of a parametrization (r(t),θ(t)) where both r(t), θ(t) are rational functions. Our study includes theoretical aspects on the shape of these curves, and algorithmic results which eventually lead to an algorithm for plotting the “interesting parts” of the curve, i.e. the parts showing the main geometrical features

Direct transformation from Cartesian into geodetic coordinates on a triaxial ellipsoid

This paper1 presents two new direct symbolic-numerical algorithms for the transformation of Cartesian co ordinates into geodetic coordinates considering the general case of a triaxial reference ellipsoid. The problem in both algorithms is reduced to finding a real positive root of a sixth degree polynomial. The first approach consists of algebraic manipulations of the equations describing the geometry of the problem and the second one uses Grobner € bases. In order to perform numerical tests and accurately compare efficiency and reliability, our algorithms together with the iterative methods presented by M. Ligas (2012) and J. Feltens (2009) have been implemented in Cþþ. The numerical tests have been accomplished by considering 10 celestial bodies, referenced in the available literature. The obtained results show that our algorithms improve the aforementioned, up-to-date reference, iterative methods, in terms of both efficiency and accuracy.Agencia Estatal de Investigación MTM2017-88796-PMinisterio de Economía, Industria y Competitividad TIN2017-86885-