TECHNIQUE OF SOLUTION OF THE OLIMPIC GEOMETRICAL PROBLEM

Abstract

The present work argues that the determination of a geometrical place of an image, equally-spaced from four incoincident points in three-dimensional space, is an important problem in the design and erection of modern architectural and engineering structures. The common issues that the problem is comprised of and essential steps of their resolution are identified. It is revealed that the essence of the problem is the contradiction between necessity of development of three-dimensional objects and two-dimensional ways of obtaining result. The purpose of the present research is to develop a way and a technique of the determination of a geometrical image, equally-spaced from four incoincident points in three-dimensional space. Objectives of the present publication are: 1. To complete the analysis of the developed solution of the Olympics geometrical problem of elevated complexity. 2. To develop a way and a technique of the graphic solution of an engineering geometrical problem of elevated complexity. The analysis of the obtained solution of the Olympics geometrical problem of elevated complexity is made. The following hypotheses are formulated: The hypothesis 1. A geometrical place of a required image, equally-spaced from four incoincident points in three-dimensional space A, B, C, D, is point K, which is equally-spaced from set points A, B, C, D if it is the center of the sphere on which all four points A, B, C, and D, non-coincident in three-dimensional space, are located. The hypothesis 2. Centre K of sphere with points A, B, C, D is located on the intersection of the middle perpendiculars set to all to all four flat sides of a pyramid, which base points A, B, C, D belong to this spherical surface. Each such middle perpendicular to a flat side of pyramid ABCD is also a geometrical place of points, equally-spaced from three corners of a corresponding side. The current research proves the validity of the hypothesis. The proposed way of the solution of the geometrical problem, consisting in graphic construction of the centers of flat sides of the pyramid, which tops settle down corners locate on sphere, construction of middle perpendiculars to each side through their centers and definitions of a point of intersection of these perpendiculars. On the basis of the proposed way, the methodology of the graphic solution of the engineering geometrical problem is developed.

Keywords: point, geometrical place, the analysis, way, technique, algorithm, the complex drawing.