Geometría diferencial de las curvas y ecuación de Riccati

"En este trabajo se examina el método de Darboux y Lie para obtener las ecuaciones paramétricas de las curvas tres y dos dimensionales a partir de la ecuación de Riccati que contiene los parámetros intrínsecos de las curvas como coeficientes en la ecuación. En este método se usa como herramienta principal el sistema de ecuaciones de Frenet-Serret que caracteriza la evolución de un triedro móvil formado por los vectores unitarios tangente, normal y binormal a lo largo de las curvas, llegando a dos ecuaciones de Riccati idénticas para dos funciones diferentes w y z que se expresan en función de las componentes del triedro móvil de Frenet-Serret. Para ilustrar el método se trabajan ejemplos de varias curvas notables tanto en dos como en tres dimensiones con distintos parámetros de curvatura y torsión.""In this thesis work, the Darboux-Lie method to obtain the parametric equation of three and two-dimensional curves starting from the Riccati equation with the intrinsic parameters of the curves as coefficients of the equation is examined. In this method, the main tool is the system of Frenet-Serret differential equations which characterizes the evolution of the moving trihedron of unit vectors of tangent, normal and binormal along the curve. It provides two identical Riccati equations for two different functions, w and z, which are related to the components of the Frenet-Serret trihedron. To illustrate the method, several examples of noted curves are worked out both in two and three dimensions, with various curvature and torsion intrinsic parameters."CONACY

Riccati equations of opposite torsions from the Lie-Darboux method for spatial curves and possible applications

A novel formulation of the Lie-Darboux method of obtaining the Riccati equations for the spatial curves in Euclidean three-dimensional space is presented. It leads to two Riccati equations that differ by the sign of torsion. The case of cylindrical helices is used as an illustrative example. Possible applications in Physics are suggested.Comment: 7 pages, 10 references, no figure