Complexes of ellipsoids with indicatrices of coordinate vectors in the form of surfaces

Abstract

The study continues in a three-dimensional affine space of complexes of three-parameter families of ellipsoids, considered earlier in a number of works by the author. A variety of ellipsoids is studied when the ends of the coordinate vectors coincide with the focal points, and the first coordi­nate straight line describes a cylindrical surface, while on the generating element there are at least three focal points that do not lie on one straight line and on one plane passing through center, and defining three conju­gate directions. A complex of ellipsoids is distinguished from the indicat­ed manifold provided that the indicatrices of the second and third coordi­nate vectors describe surfaces with tangent planes parallel to the third coordinate plane, and the end of the second coordinate vector describes a line with a tangent parallel to the first coordinate vector. An existence theorem for the variety under study is proved. The geometric properties of the complex under consideration are found. It is proved that the end of the first coordinate vector, points of the first coordinate line, and also the first coordinate plane describe a two-parameter family of planes, the end of the third coordinate vector describes a two-parameter family of cylindrical planes, a point of the third coordinate plane describes a one-parameter family of lines with tangents parallel to the first coordinate vector. The characteristic manifold of a generating element consists of six points: the vertex of the frame, three ends of the coordinate vectors, and two ends: the sum of the first and second coordinate vectors, as well as the sum of the first and third coordinate vectors. The focal manifold of the ellipsoid, the complex under study, consists of only three points, which are the ends of the coordinate vectors.