Detecting Symmetries of Rational Plane Curves

Abstract

Given a rational algebraic curve defined by means of a rational parametrization, we address here the problem of deterministically detecting whether the curve exhibits some kind of symmetry (central, mirror, rotation), and of computing the elements of the symmetry in the affirmative case. We provide effective methods for solving these questions without any conversion to implicit form. The underlying idea is the existing relationship between two proper parametrizations of a same curve, which in turn leads to algorithms where only univariate polynomials are involved. These methods have been implemented and tested in the computer algebra system Maple 15; evidence of their applicability, as well as a detailed theoretical analysis, is given.