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

Optimal reparametrizations in the square root velocity framework