GEOMETRIC MODELING OF DISCRETELY PRESENTED SURFACES TO DETERMINE THE OPTIMAL TRAJECTORIES OF TRANSPORT COMMUNICATIONS

Abstract

In the process of designing of transport communications between settlements and territorial units, it becomes necessary to determine the optimal parameters of the corresponding communications, which will ensure their minimum length. In an idealized case, the process of finding the minimum paths for laying overpasses could be reduced to searching for a geodesic line on the topographic surface of the studied relief. However, this approach is utopian, since the geodesic line envelops the surface, not taking into account the possibility that the minimum trajectory can run through unacceptable heights, lowlands, or other features of the terrain. Therefore, it is necessary to use other - more rational and controllable - approaches to determining the shortest paths for passing overpasses.

One of these approaches is the determination of minimum overpasses on the basis of specifying special discretely presented surfaces using the finite difference method or the static-geometric method of applied discrete geometry. The approach is as follows. The topographic surface under study is intersected by conditional level surfaces that form the limiting (or boundary) conditions and limit the search area for the transport connection. After that, a discrete surface is constructed on the obtained contours, which interprets the search area of ​​the overpass path. At the point (node) of the model corresponding to the destination, a concentrated load is applied, which forms uneven slopes over the entire surface. These deviations always direct search algorithms oriented on a step-by-step determination of the largest deviation at each point to the destination along the shortest path. Unfortunately, the visualization of the surfaces described above does not give a clear idea of ​​the nature of their deformation, since deformations from the application of concentrated efforts are only apparent in the areas of direct action of these efforts. In this paper, we propose a method for constructing a special additional surface that clearly demonstrates the relative conditional deformation of all its points in accordance with the initially formed surface of the slopes with an obvious distribution of the last ones.

Keywords: trajectories of transport communications, discretely presented surfaces.