Abstract
An important stage in the design of engineering networks is the development of network configurations that meet pre-defined conditions. The main thing for constructing optimization geometric models is to construct the shortest connecting lines for a given discrete set of points in space. The specified points have different weights. Such models also identify geometric factors and determine certain properties of the network configuration in the considered physical space. Geometric models define the image of the projected object and you can build various network tracing configurations that allow you to choose the optimal network. To solve this engineering problem, the article investigates and develops optimization geometric models on planes with Euclidean, orthogonal, polar metrics and algorithms. A unified algorithm for tracing extensive engineering networks on planes with different metrics has been developed. This goal is achieved by applying and generalizing the Steiner method. As you know, the Steiner problem related to the tasks of constructing a minimal spanning tree has not been solved in a general way. The practical solution to the tracing problem is to find the configuration of an extensive network with the smallest length, which has a greater impact on its cost. At the first stage of design, geometric models of engineering networks are developed, then optimization problems are solved, which are reduced to various generalizations of the problem J. Steiner [5,27]. The algorithm for solving the Steiner problem is based on dividing the given discrete points of the plane into a local subset, using the principle of least elongation and a comparative analysis of various geometric models, which allows you to select and build a network of the required configuration. A unified tracing algorithm is formulated for constructing the configuration of a Euclidean, orthogonal, and polar network. The complexity of solving the Steiner problem and engineering tracing problems is due to the fact that these problems belong to the extreme and to the class of NP-hard discrete optimization problems.