  This paper is devoted to the study of unparameterized simple curves in the
plane. We propose diverse canonical parameterizations of a 2D-curve. For
instance, the arc-length parameterization is canonical, but we consider other
natural parameterizations like the parameterization proportionnal to the
curvature of the curve. Both aforementionned parameterizations are very natural
and correspond to a natural physical movement: the arc-length parameterization
corresponds to travelling along the curve at constant speed, whereas
parameterization proportionnal to curvature corresponds to a constant-speed
moving frame. Since the curvature function of a curve is a geometric invariant
of the unparameterized curve, a parameterization using the curvature function
is a canonical parameterization. The main idea is that to any physically
meaningful stricktly increasing function is associated a natural
parameterization of 2D-curves, which gives an optimal sampling, and which can
be used to compare unparameterized curves in a efficient and pertinent way. An
application to point correspondance in medical imaging is given.
