The mapping of integer sets and Euclidean approximations

The development of discrete models for representations of nonconvex parts of $R^3$ space and the solution of routing problems with a metric that approximates the Euclidean metric on these models continue to remain fundamental in the fields of robotics, geoinformatics, computer vision, and designing of VLSI. The paper deals with a lattice-cellular model. The main attention is paid to the mapping of the integer sets $Z^2$, $Z^3$, $Z^4$ onto itself, the construction of a lattice fan under a given accuracy of metric approximation, the decomposition of equidistant graphs, and the combined application of lattice and polyhedral models for a software system of metric-topological constructions.