Kassel', Germany
Complex geometry is a synthesis of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually defines a non-closed system in which a correctly posed problem may not give a solution. Analytical geometry represents a closed system. In it, a correctly posed problem always gives solutions in the form of complex numbers, for each of which, one of the parts may be equal to zero. Finding imaginary solutions and imaginary figures formed by a set of such solutions is a new problem in descriptive geometry. Degenerated conics and quadrics, or curves and surfaces of higher orders, constitute a new class of figures and a new class of problems in descriptive geometry. For example, null-circle, null-sphere, null-cylinder, null-torus. In this paper the problem for studying the shape of second (conics, quadrics), third (conoid), and fourth (torus) order figures is posed. The latest suggest a meeting with new geometric properties of figures. Geometric operations are still immersed in the complex space E + M or real - imaginary. The examples under consideration continue a series of degenerated figures in which the null-circle splits into isotropic lines. Isotropic lines are taken as generators of surfaces. They form a cone of revolution and a hyperbolic paraboloid (an oblique plane).
analytical figure; zero-circle; imaginary addition; left isotrope; right isotrope; circular zero-cylinder; null-cylinder disintegration; isotropic planes; isotropic cone
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