Towards an Axiomatic Theory of Geodesics

Mobi spaces provide an algebraic abstraction of interpolation between points. In this paper we investigate their relation with geodesic structures. We present a construction of mobi spaces from a parametrized map, formulate conditions under which the mobi axioms follow from a given parametrization, and show that a broad class of second-order differential equations naturally gives rise to mobi spaces. This establishes a general framework linking geodesic motion and mobi structures. As an illustration, we obtain an explicit mobi space associated with a metric of non-constant negative curvature. Taken together, these results suggest that mobi spaces may provide a useful axiomatic framework for the study of geodesic interpolation.