Building a Sphere from Imaginary Points

Abstract

Euclidean spaces of various dimensions do not contain imaginary images and objects by definition, but are inextricably linked with them through special cases, and this leads to the need to expand the field in geometry into the region of imaginary values [1, 19, 26]. Such an extension, i.e. adding to the field of real coordinates spaces of different dimensions, the field of imaginary coordinates leads to different variants of spaces of different dimensions, depending on the chosen axiomatics. Earlier in a number of articles, examples of solving some actual problems of geometry using imaginary geometric images and objects were shown [4, 5, 6, 13, 21, 22, 29]. 
 The article provides constructions for constructing a sphere from four predetermined points, of which one pair or both pairs of points can be imaginary complex conjugate. The construction is carried out on combined diagrams by the methods of descriptive geometry by analogy with the well-known problem of constructing a sphere from four real points. The construction of a sphere is based on seven auxiliary constructions for constructing a circle from points that can be imaginary conjugates. Both 3D problems of constructing spheres for given points and methods of 2D construction problems for determining the required imaginary points are considered. A method for calculating the parameters of the obtained sphere is described. The application of the method to other problems of descriptive geometry, for example, to the problems of finding geometric places of points, is considered. equidistant from two given surfaces. Recently, this issue has been intensively studied, for example, in the works [5, 6].