INTERACTION OF COMPLEX GEOMETRIC OBJECTS WITH VARIABLE METRIC CHARACTERISTICS

Abstract

The interaction of material objects involved in the synthesis of complex technical systems requires consideration of their spatial shape, metric characteristics, as well as restrictions on their location. In the synthesis of complex systems there are problems related to the theory of geometric design, ie problems of optimization of placement (of covering, partitioning, motion modeling) and which are related to mathematical and geometric modeling of objects and their relationships.
 Despite the existence of various models and methods of solving problems of geometric design, they are still relevant in those areas whose formalization is insufficient for the application of existing models and methods, due to the need to take into account the characteristics of each subject area. This, in turn, leads to the need to build new models, formulate new tasks and develop effective methods for solving them.
 Within the class of placement problems, the paper considers the problem of rational placement of complex objects with variable metric characteristics, as a result of which the configuration of placement of objects of new spatial forms is synthesized. Therefore, the urgent problem is the further development of the mathematical apparatus for the synthesis and description of complex objects and the conditions of their mutual nonintersection.
 To do this, the model of complex objects is built, which consists of the one main and a number of auxiliary. The main object can rotate continuously, and the auxiliary ones have the ability to rotate continuously relative to the specified common points with the main in a given range of angles (relative to the angle of rotation of the main object). Analytical expressions are obtained for the conditions of non-intersection of complex objects considered in the paper.
 The mathematical apparatus of interaction of geometric objects is the basis of methods for modeling of placevent given constraints, modeling the movement of people.
 As an example, the paper presents a partial case, namely the three-component model, which is a horizontal projection of the human body, and the result of solving the problem of determining the maximum number of objects placed in a rectangular area by selecting objects according to a given sequence of numbers from some set.