On the Peculiarities of the Constructive Solution For Dandelin Spheres Problem

Abstract

This paper is devoted to analysis of Dandelin spheres problem based on the constructive geometric approach. In the paper it has been demonstrated that the traditional approach used to this problem solving leads to obtaining for only a limited set of heterogeneous solutions. Consideration of the problem in the context of plane and space’s projective properties by structural geometry’s methods allows interpret this problem’s results in a new way. In the paper it has been demonstrated that the solved problem has a purely projective nature and can be solved by a unified method, which is impossible to achieve if conduct reasoning and construct proofs only on affine geometry’s positions. The research’s scientific novelty is the discovery and theoretical justification of a new classification feature allowing classify as Dandelin spheres the set of spheres pairs with imaginary tangents to the quadric, as well as pairs of imaginary spheres with a unified principle for constructive interrelation of images, along with real solutions. The work’s practical significance lies in the extension of application areas for geometric modeling’s constructive methods to the solution of problems, in the impro vement of geometric theory, in the development of system for geometric modeling Simplex’s functional capabilities for tasks of objects and processes design automation. The algorithms presented in the paper demonstrate the deep projective nature and interrelation of such problems as Apollonius circles and spheres one, Dandelin spheres one and others, as well as lay the groundwork for researches in the direction of these problems’ multidimensional interpretations. The problem solution can be useful for second-order curves’ blending function realization by means of circles with a view to improve the tools of CAD-systems’ design automation without use of mathematical numerical methods for these purposes.