
12(85) 2015 MATHEMATICS
М.А. Cheshkova
ОneSided Surfaces
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 onesided surface. The Mebius band is the onesided surface. Crosscap, Roman surface, Boy surface, Klein bottle are also onesided surfaces. We define the closed curve on the torus using 4π periodic vectorfunction ρ = ρ(v). Then vectorfunction s(v) = ½ (ρ(v) + ρ(v + 2π)) is a 2πperiodic vectorfunction, and the function l(v) = ½ (ρ(v) − ρ(v + 2π)) is a 2πantiperiodic vectorfunction. The equations of the Mobius band, Klein bottle and crosscap 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)1.230
Key words: Klein bottle, Mobius band, crosscap, 2πperiodic function, 2πantiperiodic function
