Methods of forming orthogonal polyhedra for cutting and packing objects of complex geometry

The article deals with the problem of packing objects of arbitrary geometry. Modern methods of designing irregular packing schemes use a mathematical model based on phi-functions and a hodograph vector function of dense placement. These methods make it possible to obtain exact solutions, but they are time-consuming and very sensitive to the dimension of the problem being solved and the degree of detail of the geometry of vector objects. The use of a discrete representation of placed objects in the form of orthogonal polyhedra can signifi increase the speed of construction a packing, which makes the problem of adequately transforming the shape of placed objects (vector models in the two-dimensional case and polygonal models in the three- dimensional case) relevant. The aim of the study is to systematize methods that provide the formation of orthogonal polyhedra of various dimensions for describing objects and containers of arbitrary geometry. Methods for creating orthogonal polyhedra based on set-theoretic operations (addition, subtraction and intersection), analytical modeling using a set of functions and relational operators, as well as voxelization of fl and volumetric object models are considered. The use of set-theoretic operations is best suited for the manual creation of orthogonal polyhedra with relatively simple geometry. The method of analytical modeling is intended for the formation of voxelized objects based on geometric fi es described by a set of analytically specifi functions. The application of various relational operators to obtain orthogonal polyhedra that describe the contour, internal and external regions of analytical given objects is shown. An algorithm for creating a container in the form of an orthogonal polyhedron based on a given vector model is proposed, which makes it possible to solve problems of irregular packing of objects inside containers of arbitrary shape. All the methods presented in the article are programmatically implemented with a generalization in terms of dimension and are applicable to solving any types of cutting and packing problems.