Оne-Sided Surfaces
Abstract
Let a normal vector be defined along a closed curve on surfaces. If you return to the starting point, and the normal direction coincides with the original one, regardless of the choice of the curve, then the surface is called bilateral. Otherwise, we have a one-sided surface. The Mebius band is the one-sided surface. Cross-cap, Roman surface, Boy surface, Klein bottle are also one-sided surfaces. We define the closed curve on the torus using 4π- periodic vector-function ρ = ρ(v). Then vectorfunction s(v) = ½ (ρ(v) + ρ(v + 2π)) is a 2π-periodic vector-function, and the function l(v) = ½ (ρ(v) − ρ(v + 2π)) is a 2π-antiperiodic vector-function. The equations of the Mobius band, Klein bottle and cross-cap are defined using the obtained functions. A normal vector is defined along a closed curve s = s(v). If you return to the starting point then the normal direction is opposite to the original one. Examples of these surfaces are constructed using a mathematical software package.
DOI 10.14258/izvasu(2015)1.2-30
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