We propose a geometric method for quantifying the difference between
parametrized curves in Euclidean space by introducing a distance function on
the space of parametrized curves up to rigid transformations (rotations and
translations). Given two curves, the distance between them is defined as the
infimum of an energy functional which, roughly speaking, measures the extent to
which the jet field of the first curve needs to be rotated to match up with the
jet field of the second curve. We show that this energy functional attains a
global minimum on the appropriate function space, and we derive a set of
first-order ODEs for the minimizer.