Conformal and geodesic mappings of generalized equidistant spaces

We consider conformal and geodesic mappings of generalized equidistant spaces. We prove the existence of mentioned nontrivial mappings and construct examples of conformal and geodesic mapping of a 3-dimensional generalized equidistant space. Also, we find some invariant objects (three tensors and four which are not tensors) of special geodesic mapping of generalized equidistant space.

The Extension of Geodesics

We introduced an extension of traditional geodesic. Local neighbourhood of smooth curve on manifold shows good propositions which make it possible to construct a geodesic mapping from the neighbourhood to a rectangular district of UV-plane. Length of subsurface extended from smooth curve is defined according to the parameterization induced by geodesic mapping. And extended geodesic is the curve with minimal length of subsurface it extends. We also proposed a constrained mass-spring based approach to solve extended geodesic on discrete mesh. Experimental result shows that it is a high precision approximation for problems of measurement where the path can not be reduced as a single curve.