  This chapter reviews some past and recent developments in shape comparison
and analysis of curves based on the computation of intrinsic Riemannian metrics
on the space of curves modulo shape-preserving transformations. We summarize
the general construction and theoretical properties of quotient elastic metrics
for Euclidean as well as non-Euclidean curves before considering the special
case of the square root velocity metric for which the expression of the
resulting distance simplifies through a particular transformation. We then
examine different numerical approaches that have been proposed to estimate such
distances in practice and in particular to quotient out curve reparametrization
in the resulting minimization problems.
