Особенность отображения гиперповерхности четырехмерного пространства

Abstract

Investigation of singularity related to a mapping by the orthogonal projection in a four-dimensional space of a hypersurface given by parametric equations is presented in this paper. On this investigation’s basis have been proposed in a united way three approaches to the hypersurface’s discriminant determination. Thus, have been defined conditions which the investigated hypersurface’s discriminant set and criminant are satisfied. Have been obtained dependencies settling the relationship between parameters of the hypersurface at its discriminant points. They are used to determine the singularity of the hypersurface mapping by analytical methods in general form. The complexity of this approach (the first one) is that an equation connecting the hypersurface’s parameters contains its differential characteristics, and often is the transcendental one in applications, that causes certain difficulties when solving it. Have been obtained dependences in carrying out of which the hypersurface’s discriminant has an edge of regression. A study of hypersurface sections by hyperplanes which are parallel to coordinate hyperplanes has been performed. The last ones contain a coordinate axis along which the hypersurface mapping is performed. It has been established that curves obtained in these sections have extreme points belonging to the hypersurface’s discriminant. Such property is used to calculate the points of the hypersurface’s discriminant by numerical methods without using the hypersurface’s differential characteristics, and it is a basis for the second approach to solving the problem posed. It has been also demonstrated the use of 3D modeling for study of hypersurface’s different sections, as well as its discriminant that represents the third approach to the study. All three approaches having a common basis can be used both independently and complement each other in determining the envelope for one-parameter family of surfaces. As an example has been considered a hypersurface formed by a family of spheres. Based on stated results have been obtained equations determining the hypersurface’s discriminant and this family’s corresponding envelope, as well as various sections. These equations have been used for creating of polygonal 3D models of the hypersurface’s discriminant, and some of the hypersurface’s sections.