An extension of the class of lattice graphs is considered. In order to become close to the Euclidean metric, inclusion of additional ribs with weights equal to the corresponding lengths of vectors in the Euclidean space is performed in a neighborhood on the lattice. A correspondence between the coordinates of nodes incident to the additional ribs and sequences of irreducible Farey-Cauchy fractions is established. An algorithm for constructing a set of the shortest paths on such a lattice is proposed. In essence, this algorithm models the "wave" process of constructing the field of all shortest paths from a set-source. Some estimates and examples are given to illustrate the computer realization of the algorithm proposed.