Kassel', Germany
Complex geometry consists of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually determines an open system in which a correctly posed problem may give no solution. Analytical geometry is an example of a closed system, in which the previously mentioned problem always gives a solution as a complex number, whose one of the parts may turn out to be zero. Development of imaginary solutions and imaginary figures is a new task for descriptive geometry. Degenerated conics and quadrics set up a new class of figures and a new class of descriptive geometry’s problems. For example, a null circle, null sphere, null cylinder, and a cone as a hyperboloid degenerated to an asymptote. The last ones necessarily lead to imaginary solutions in geometric operations. In this paper it has been shown that theorems formulated in one geometry are also valid in conjugate geometry as well, while the same figures of conjugated geometries visually look different. So imaginary points exist only by pairs, the imaginary circle is not round one, the centers of dissimilar circles’ similarity do not belong to the centerline and other examples. For solution, a number of problems on geometric relations, and operations with degenerated conics and quadrics, as well as several problems from 4D-geometry are proposed. Solutions for above mentioned problems are given in section 9. In this paper some examples of new problems for descriptive geometry have been considered. It has been shown that the new problems require access to a complex space. New figures consist of two parts, a real figure and a figure of its imaginary complement.
analytical figure, synthetic figure, imaginary sphere, imaginary cylinder, degenerated surfaces, null sphere, null cylinder, imaginary complement, isotropes, contact, intersection, 4D figures
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