  We study the pull-back of the 2-parameter family of quotient elastic metrics
introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized
loops. This point of view has the advantage of concentrating on the manifold of
arc-length parameterized curves, which is a very natural manifold when the
analysis of un-parameterized curves is concerned, pushing aside the tricky
quotient procedure detailed in Lahiri-Robinson-Klassen of the preshape space of
parameterized curves by the reparameterization (semi-)group. In order to study
the problem of finding geodesics between two given arc-length parameterized
loops under these quotient elastic metrics, we give a precise computation of
the gradient of the energy functional in the smooth case as well as a
discretization of it, and implement a path-straightening method. This allows us
to have a better understanding of how the landscape of the energy functional
varies with respect to the parameters.
