  Given a planar curve defined by means of a real rational parametrization, we
prove that the affine values of the parameter generating the real singularities
of the offset are real roots of a univariate polynomial that can be derived
from the parametrization of the original curve, without computing or making use
of the implicit equation of the offset. By using this result, a finite set
containing all the real singularities of the offset, and in particular all the
real self-intersections of the offset, can be computed. We also report on
experiments carried out in the computer algebra system Maple, showing the
efficiency of the algorithm for moderate degrees.
