Visualization of multidimensional areas of solutions to mathematical models of complex objects

Abstract

Visualization of mathematical models of multidimensional objects implicitly defined as a surface F (X) = 0 is performed using two- and three-dimensional projections obtained by rotating and shifting coordinates. In the new coordinates, the model has the form y = f(x) where one of the variables is dependent (this is the direction of design), and the rest make up the vector of independent variables. A new model is often more convenient than the original one, but not all design directions adequately reflect its properties since it is possible to transfer projections of several points of the original surface into one (gluing). For some directions (directions of ambiguity), the projection may consist of points with several preimages (corresponds to the ambiguity of the function f). In this case, the projection is divided into the area of ambiguity and the area of unambiguity. For other directions (directions of uniqueness), the entire projection is the domain of uniqueness. The paper investigates the evolution of design directions and areas of ambiguity and unambiguity of projections of complex objects. A criterion for choosing the direction of unambiguity is proposed in which all points of the model remain different, i.e., multiplicity visually disappears (“hidden multiplicity”). Examples of applying the criterion for models of objects of various geometries are given.