Efficient detection of symmetries of polynomially parametrized curves

We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on an algebraic relationship between proper parametrizations of a same curve, which leads to a triangular polynomial system that can be solved in a very fast way; in particular, curves parametrized by polynomials of serious degrees/coefficients can be analyzed in a few seconds. In our analysis we provide a good number of theoretical results on symmetries of polynomial curves, algorithms for detecting rotation and mirror symmetry, and closed formulae to determine the symmetry center and the symmetry axis, when they exist. Some observations and empiric results for the case of polynomial parametrizations with floating point coefficients are also reported.