CONDITIONS OF NON-INTERSECTION ELLIPSES TAKING INTO ACCOUNT THE MAXIMUM ALLOWABLE DISTANCES

Abstract

The informatization of modern society, of which Ukraine is an integral part, and the widespread use of computers have led to a significant impetus in the development of mathematical modeling. This development contributes to the creation of models, the geometric properties of which make it possible to develop methods for modeling the placement, movement of objects of any nature according to specified constraints. One of the urgent problems is the development of scientifically based plans for the evacuation of people, the main components of which are programs for modeling the movement of the human flow. Therefore, an urgent problem is the development of models and methods for modeling the movement of human flows, for which there is a need to develop programs that would adequately reflect the real processes of their movement.

The problem of modeling the movement of people at each discrete moment of time is a configuration of the placement of objects. In the applied problem considered in this article, the object of placement (movement) is a person. It has been shown in the literature that with the free category of motion, the most adequate model of the projection of the human body onto a horizontal plane is an ellipse. It should be noted that the problem of modeling the movement of people is characterized by the presence of placement restrictions, the main of which are the conditions of non-intersection, and additional restrictions, among which it is possible to determine the orientation of objects, restrictions on the maneuverability and comfort of movement, etc.

In practice, the problem often arises of modeling the movement of people in groups, examples of which are family members or rescuers of one unit. The distance between people of each group should not exceed the maximum allowable. Taking into account the maximum allowable distances between objects allows you to combine them into subgroups, and the specified maximum distances between subgroups allows you to combine them into groups. Therefore, an urgent task is to simulate the conditions of interaction between objects, taking into account the maximum permissible distances between them. The work formalized the listed restrictions on the interaction of objects. For the analytical description of the conditions of non-intersection of objects, in particular ellipses, quasi-phi-functions for ellipses have been modified, observing the maximum allowable distances between them.

Keywords: mathematical modeling, modeling of human movement, conditions for non-intersection of ellipses, maximum allowable distances