Constructive algorithms for automated solutions to positional problems

This paper presents the theoretical foundations for an approach to automated solutions of positional problems involving intersections of geometric objects in Euclidean space. A conceptual framework is developed based on concepts and definitions related to the equivalence relation known in set theory. This framework includes: an equivalence relation defined on the set of geometric objects in Euclidean space; the dimension and factorization of the object set, represented as a collection of equivalence classes; and the projected factor set of space. A set-theoretic basis for representing the projection operation is provided, enabling the development of constructive algorithms to solve positional problems involving intersections of spatial geometric objects. These algorithms have been implemented within the KOMPAS-3D CAD environment. The theoretical foundations and analysis of the algorithm functionality establish definitions for "geometric modeling" and "3D modeling." The proposed approach to automating positional problem solutions essentially constitutes a 3D model of these solutions through the creation of a virtual model. The paper includes examples of automated 3D solutions to intersection problems, descriptions of the algorithms used in their design, and examples of practical applications of these algorithms.