Un problema de David A. Singer sobre curvas planas

Abstract

Se estudia el caso particular concreto del llamado Problema de Singer sobre curvas planas, consistente en determinar aquellas curvas cuya curvatura depende de la distancia a una recta. Se describen cinco familias específicas cuyas representaciones analíticas pueden expresarse en términos de funciones elementales, a partir de un resultado abstracto de integración en la línea del resultado principal del propio Singer. De todas las curvas estudiadas, se consigue también su ecuación intrínseca y se realiza su representación gráfica.

A concrete particular case of the so-called Singer Problem on planar curves is studied, consisting in determining those curves whose curvature depends on the distance to a line. Five specific families are described whose analytical representations can be expressed in terms of elementary functions, based on an abstract result of integration following the spirit of the main result of Singer himself. Among all the studied curves, their intrinsic equations are also achieved and their graphic representations are made.