We propose a notion of distance between two parametrized planar curves,
called their discrepancy, and defined intuitively as the minimal amount of
deformation needed to deform the source curve into the target curve. A precise
definition of discrepancy is given as follows. A curve of transformations in
the special Euclidean group SE(2) is said to be admissible if it maps the
source curve to the target curve under the point-wise action of SE(2) on the
plane. After endowing the group SE(2) with a left-invariant metric, we define a
relative geodesic in SE(2) to be a critical point of the energy functional
associated to the metric, over all admissible curves. The discrepancy is then
defined as the value of the energy of the minimizing relative geodesic. In the
first part of the paper, we derive a scalar ODE which is a necessary condition
for a curve in SE(2) to be a relative geodesic, and we discuss some of the
properties of the discrepancy. In the second part of the paper, we consider
discrete curves, and by means of a variational principle, we derive a system of
discrete equations for the relative geodesics. We finish with several examples
