QUASI-PHI-FUNCTIONS FOR ANALYTICAL DESCRIPTION CONDITIONS OF NON-INTERSECTIONS OF COMPOUND OBJECTS IN THE PROBLEMS OF PLACING AND MODELING MOVEMENT OF MOVEMENTS OF PEOPLE

Abstract

The problems of optimal placement of objects are the subject of study of computational geometry, and methods of their solution are the direction of the theory of operations research. The placement tasks include packing and cutting tasks, which have a wide range of scientific and practical applications in powder metallurgy, in the mining industry for modeling the movement of bulk substances, at the analysis of structures of liquids and glass, in the problems of logistics for modeling of the optimal packing of goods, , in the problems of modeling individual movement people during their evacuation from buildings, etc. From the point of view of modeling methods, the above classes of application problems belong to the class of geometric design problems with a specific constraint system that is related to their geometric properties. This class of problems belongs to the class of NP-complexes, which are usually solved by heuristic algorithms. In order to develop efficient algorithms based on local and global optimization methods, it is necessary to build adequate mathematical models based on the analytical representation of the conditions of nonintersection of objects, taking into account their continuous translations and rotations. One of the actual problem nowadays is the task of simulating the movement of the flow of people, which, at any given time, can be considered as placing people for specified constraints. The results of the analysis show that there is no model of individually current movement of people that is adequate to the real flow. When moving people in the stream are the following categories of motion: comfortable, calm, active, of high activity. When the movement category moves into the category of active movement with possible force actions, the density of flow increases, which leads to natural deformations of the human body. The paper proposes a model of the human body, taking into account its natural deformities, as a three-component model, which is a union of three non-rigidly bound ellipses. The main ellipse rotates continuously within the maneuverability angle relative to the main direction of movement, and the auxiliary ellipses can rotate within acceptable limits relative to the main one.For the analytical description of the conditions of non-intersections of threecomponent objects, a modification of the quasi-phi-functions of compound objects is proposed, which is the basis of algorithms for modeling the current active movement of people and the packing of the considered class of objects.