Klein Bottle Hermann Karcher Parametric Equations for the Twisting Klein Bottle x = (aa + cos(v / 2) * sin(u) - sin(v / 2) * sin(2 * u)) * cos(v) y = (aa + cos(v / 2) * sin(u) - sin(v / 2) * sin(2 * u)) * sin(v) z = sin(v / 2) * sin(u) + cos(v / 2) * sin(2 * u) See the Mobius Strip first. The non-orientable surfaces are "one-sided", and this concept can best be understood if one starts from a Mobius Strip. Imagine that we modify a Torus by rotating a figure-eight instead of a circle. If we color the two sides of this surface differently then one loop of the figure-eight has one color, the other loop the other color. (One can make such a surface in the Surface Category, by taking the Lemniscate as the meridian curve after choosing "User Defined (Rotation)" from the Surface menu. (This is in fact is the default meridian. ) The default Klein Bottle is obtained by one further modification: Rotate the meridian figure-eight in its plane by 180 degrees as the curve is being rotated 360 degrees about the axis of rotation. One can see the figure-eight better if in "Set u,v ranges" one sets vmin = 0.5. Use ``Distinguish Sides by Color'' in this cut open view. One can see this in a morph with aa= 3, umin=0, umax=2 * π, vmin=0, 0.5 < umax < 2 * π. The default morph starts from the Mobius Strip -0.4 < u < 0.4, 0 < v < 2 * π, aa = 3 and increases the width of the Mobius Strip until it closes to the Klein Bottle at - π < u < π. There are in fact three different kinds of Klein Bottles which cannot be deformed into each other: (i) The present one, where the figure-eight rotates to the left in its plane; (ii) The mirror image of the present one, where the figure-eight rotates clockwise; (iii) a Klein Bottle with mirror symmetry, glass models are frequently made.
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Moebius_Strip and Klein_Bottle.pdf