  This paper addresses the problem of determining the symmetries of a plane or
space curve defined by a rational parametrization. We provide effective methods
to compute the involution and rotation symmetries for the planar case. As for
space curves, our method finds the involutions in all cases, and all the
rotation symmetries in the particular case of Pythagorean-hodograph curves. Our
algorithms solve these problems without converting to implicit form. Instead,
we make use of a relationship between two proper parametrizations of the same
curve, which leads to algorithms that involve only univariate polynomials.
These algorithms have been implemented and tested in the Sage system.
