Danilin O. Geometric modeling of placement of non-oriented ellipses according to the given constraints

The object of the study is the process of placing non-oriented ellipses. The aim of the study is to increase the efficiency of solving the problems of placement of ellipses under the specified constraints by developing constructive means of geometric modeling of relations between ellipses (touch, nonintersection, intersection) and optimizing their placement. During researches methods of mathematical modeling, analytical, computational geometry, geometric modeling, geometrical design, optimization, system approach, elements of topology were used. In this article, the problem of packing of sets of ellipses in a given region taking into account conditions of nonintersection and technological restraints which are concretized in the conditions of the applied problem is formulated. The model of packing of a set of ellipses in a rectangle of minimum dimensions is improved. Continuous ellipse rotations and translations are allowed, the possibility of availability of minimum admissible distances between them is assumed. New quasi-phi-functions are constructed for modeling of the relations of ellipse nonintersection and to define belonging of an ellipse to the container. An effective algorithm for locally optimal solutions is proposed in terms of its complexity. The model of the individual-and-flow movement of individuals approximated by ellipses with specification of technological restraints is constructed. The method of geometric modeling of the movement of people based on the criterion of the maximum of their total displacement is given. The use of this method is advisable when choosing rational evacuation routes at objects of different designation. The results of the work were introduced into the work of the Main Department of the State Service of Ukraine for Emergencies in the Kharkiv region.