THE DUALITY PRINCIPLE IS THE THEORETICAL BASIS OF INTERRELATION OF SYNTHETIC AND ANALYTICAL METHODS OF SOLVING GEOMETRIC PROBLEMS  Abstract (English):
There is a tendency of dividing of the teachers of the engineering graphics departments into three groups: 1) conservatives: we studied the descriptive geometry by textbook of V.O. Gordon, it is supposed to be the textbook for our students; computer graphics must be preceded by "manual" engineering graphics; 2) radicals: the descriptive geometry as an academic discipline has run dry; the training on the basis of 3D modeling is up to date; 3) moderate specialists: descriptive geometry, as a part of the integrated course of engineering geometry, along with engineering and computer graphics, is aimed at providing related branches of mathematics and General engineering disciplines. The article is devoted to justify the positions of specialists. The transformation of the traditional course of descriptive geometry in the engineering geometry can be provided in case of: • considering jointly the synthetics and analytical methods to solve geometric problems, • improving the subject of the course with the forms of the multidimensional space. Theoretical basis of this approach is the principle of duality in multidimensional projective space. It is shown on a geometric interpretation of the solution of a system of n linear equations with n unknowns. The interrelation of synthetic and analytical methods is shown on the examples of setting linear forms of four-dimensional space (point A, straight line a, 2-plane α2 (ABC), 3-plane α3 (ABCD)) general provisions, projecting figures and the solutions of position tasks on the affiliation and the intersection shown. Noted that any positional problem is reduced to solving p + 1 linear systems of n equations with n unknowns, where P is the dimension of the desired p-plane αp. Therefore, following the principle of learning from simple to complex, methodologically correctly consistently explain the algorithms of constructing of a point, line, ..., p-plane. This approach becomes mandatory for entry • from linear to nonlinear forms; • from the solution of educational problems involving simple lines and surfaces to the solution of applied problems involving compound curves and surfaces (one-dimensional, two-dimensional and multidimensional contours); • to the solution of optimization problems methods of geometric programming.

Keywords:
descriptive geometry, synthetic and analytical methods of solving geometric problems, dimension, the degree of freedom, the principle of duality in multidimensional projective space.

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