Reparameterization invariant distance on the space of curves in the hyperbolic plane

We focus on the study of time-varying paths in the two-dimensional hyperbolic space, and our aim is to define a reparameterization invariant distance on the space of such paths. We adapt the geodesical distance on the space of parameterized plane curves given by Bauer et al. in [1] to the space Imm([0, 1],H) of parameterized curves in the hyperbolic plane. We present a definition which enables to evaluate the difference between two curves, and show that it satisfies the three properties of a metric. Unlike the distance of Bauer et al., the distance obtained takes into account the positions of the curves, and not only their shapes and parameterizations, by including the distance between their origins.