Images of Linear Conditions on a Manhattan Plane

In this paper are considered planar point sets generated by linear conditions, which are realized in rectangular or Manhattan metric. Linear conditions are those expressed by the finite sum of the products of distances by numerical coefficients. Finite sets of points and lines are considered as figures defining linear conditions. It has been shown that linear conditions can be defined relative to other planar figures: lines, polygons, etc. The design solutions of the following general geometric problem are considered: for a finite set of figures (points, line segments, polygons...) specified on a plane with a rectangular metric, which are in a common position, it is necessary to construct sets that satisfy any linear condition. The problems in which the given sets are point and segment ones have been considered in detail, and linear conditions are represented as a sum or as relations of distances. It is proved that solution result can be isolated points, broken lines, and areas on the plane. Sets of broken lines satisfying the given conditions form families of isolines for the given condition. An algorithm for building isoline families is presented. The algorithm is based on the Hanan lattice construction and the isolines behavior in each node and each sub-region of the lattice. For isoline families defined by conditions for relation of distances, some of their properties allowing accelerate their construction process are proved. As an example for application of the described theory, the problem of plane partition into regions corresponding to a given set of points, lines and other figures is considered. The problem is generalized problem of Voronoi diagram construction, and considered in general formulation. It means the next: 1) the problem is considered in rectangular metric; 2) all given points may be integrated in various figures – separate points, line segments, triangles, quadrangles etc.; 3) the Voronoi diagram’s property of proximity is changed for property of proportionality. Have been represented examples for plane partition into regions, determined by two-point sets.