Editing Möbius strip (section)

===Sweeping a line segment===
{{CSS image crop
|Image = Mobius strip.gif
|bSize = 400
|cWidth = 185
|cHeight = 150
|oTop = 115
|oLeft = 105
|Description=A Möbius strip swept out by a rotating line segment in a rotating plane}}
{{CSS image crop
|Image=Plucker's conoid (n=2).gif
|bSize=360
|cWidth=240
|cHeight=240
|oTop=60
|oLeft=60
|Description=[[Plücker's conoid]] swept out by a different motion of 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 {{nowrap|lines.{{r|maschke}}}} 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,
<math display=block>
\begin{align}
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{align}</math>
for <math>0 \le u< 2\pi</math> and {{nowrap|<math>-1 \le v\le 1</math>,}}
where one parameter <math>u</math> describes the rotation angle of the plane around its central axis and the other parameter {{nowrap|<math>v</math>}} 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 <math>xy</math>-plane and is centered at {{nowrap|<math>(0, 0, 0)</math>.{{r|parameterization}}}} 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 {{nowrap|connected.{{r|split-tori}}}}

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 {{nowrap|strip.{{r|francis}}}} It has applications in the design of {{nowrap|[[gear]]s.{{r|dooner-seirig}}}}