Möbius strip
Sweeping a line segment

One way to embed the Möbius strip in three-dimensional Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of its lines. For the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity of the plane's rotation. This can be described as a parametric surface defined by equations for the Cartesian coordinates of its points, x (u, v) = (1 + v 2 cos u 2) cos u y (u, v) = (1 + v 2 cos u 2) sin u z (u, v) = v 2 sin u 2 {\displaystyle {\begin{aligned}x(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\cos u\\y(u,v)&=\left(1+{\frac {v}{2}}\cos {\frac {u}{2}}\right)\sin u\\z(u,v)&={\frac {v}{2}}\sin {\frac {u}{2}}\\\end{aligned}}} for 0 ≤ u < 2 π {\displaystyle 0\leq u<2\pi } and − 1 ≤ v ≤ 1 {\displaystyle -1\leq v\leq 1}, where one parameter u {\displaystyle u} describes the rotation angle of the plane around its central axis and the other parameter v {\displaystyle v} describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the x y {\displaystyle xy} -plane and is centered at (0, 0, 0) {\displaystyle (0,0,0)}. The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in the plane that it rotates within, and the Möbius strip that it generates forms a slice through the solid torus swept out by this disk. Because of the one-sidedness of this slice, the sliced torus remains connected.

A line or line segment swept in a different motion, rotating in a horizontal plane around the origin as it moves up and down, forms Plücker's conoid or cylindroid, an algebraic ruled surface in the form of a self-crossing Möbius strip. It has applications in the design of gears.
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