Isoline families in the space with rectilinear distances

The paper is devoted to planar linear sets generated by linear conditions which are realized for rectangular or Manhattan metrics. The aim of this paper is to give an approach to researching the properties of linear conditions defined on Manhattan plane. Linear conditions are written as a finite sum of products Manhattan distances and real numerical factors. The paper presents a constructive method to solve the next general geometric problem. Let finite sets of linear figures (segments of line, polygons) are given on Manhattan plane at general positions. Find a planar set corresponding to given linear condition based on the given sets. The set of broken lines corresponding to given conditions generates a family of isolines for given sets. Constructive algorithm to build the family is suggested. The algorithm is based on the rectangular Hanan lattice and calculation numerical values in each cell. Some theorems applied to these cases are formulated in the paper.