Singularity Features Identification in Spatial Problems of Transport Logistics

The objective of this paper is to study the problem of constructing an optimal route in three-dimensional space to a target set, which is an object of nontrivial geometry. The authors developed algorithms for constructing layers with specified values of the cost function for moving the dynamic body (apparatus) to a target set. It is hypothesized that the dynamic body moves for significant distances and that its dimensions are negligible; this allows for the modeling of the body as a material point. This paper sets forth a distinctive configuration: that of a bisector on which the cost function becomes singular, losing smoothness. The bisector is constituted by specific points, from which multiple optimal trajectories emerge for each point on the bisector. The necessary and sufficient conditions for the identification of an optimal trajectory for some segment are obtained. The formulas for the extreme points of a singular set in terms of the curvature of the surface of the target set were derived. The paper provides an example of constructing cost function level surfaces based on bisector extraction.