To the Geometry of the Klein Bottle
УДК 514.756
Abstract
This paper continues the author’s series of works on the modeling of one-sided surfaces. There is a closed curve (disorienting contour) on a one-sided surface, and this curve has the property of changing the sign when it is bypassed local orientation in tangent space.
A one-sided surface is a Klein bottle. Two smooth vector functions are considered. It is assumed that one of them is 2 pi — periodic, the other is 2 pi — antiperiodic. The Klein bottle equations, disorienting contours, and the equations of two Mobius sheets into which a Klein bottle is cut are determined using the obtained functions. In this paper, the inversion of the Klein bottle is investigated.
We prove that if a Klein bottle does not pass through the center of inversion, then the inversion of the Klein bottle is also the Klein bottle. It is also proved that if the Klein bottle does not pass through the center of inversion, then the disorienting contours of the Klein bottle under the inversion will pass into the disorienting contours. Models of the considered surfaces are built with the help of the computer mathematics system.
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