An Inversion of a Mobius band
Abstract
If a closed curve (disorienting contour) exists the surface in E3, and the local orientation in the tangent space changes its sign while tracing this curve, then the surface is called a one-sided surface. A Mobius band is the one-sided surface. Two smooth vector-functions are considered in the Euclidian space E3: s = s(u), l = l(u), u ∈ [-n, п]. It is assumed that s = s(u) is a 2п -periodic vector-function, and l = l(u) is a 2п -antiperiodic vector-function. Equations for the Mobius band and the disorienting contours are found using the obtained functions. An inversion of Mobius band is studied. It is proved that if the Mobius band does not cross through the center of inversion, then the inversion of the Mobius band is a Mobius band. It is also proved that if the Mobius band does not cross through the center of inversion, then the disorienting contours of the Moebius band transit to disorienting contours also. An example of the Mobius band is considered. The closed curve is defined on a torus using 4п -periodic vector-function р = p(v). Then vector-function s(v) = 1/2 (p(v) + p(v + 2п)) is a 2п-periodic vector-function and the function l(v) = 1/2 (p(v)-p(v+2n)) is a 2п -antiperiodic vector-function. Equations of the Mobius band and its inversion are developed using the obtained functions. Examples of these surfaces are constructed with mathematical software packages.
DOI 10.14258/izvasu(2017)4-29
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