  The square root velocity framework is a method in shape analysis to define a
distance between curves and functional data. Identifying two curves if they
differ by a reparametrisation leads to the quotient space of unparametrised
curves. In this paper we study analytical and topological aspects of this
construction for the class of absolutely continuous curves. We show that the
square root velocity transform is a homeomorphism and that the action of the
reparametrisation semigroup is continuous. We also show that given two
$C^1$-curves, there exist optimal reparametrisations realising the minimal
distance between the unparametrised curves represented by them. Furthermore we
give an example of two Lipschitz curves, for which no pair of optimal
reparametrisations exists.
