TITLE: Understanding the Equation of a Möbius Strip
QUESTION [22 upvotes]: I am in HL Math and trying to finish my IA.  My topic is the Möbius band.  The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-jargon filled explanation of parametric equations that confuse me.  Is there anyone who can explain how to derive the equations/ what they literally mean?  It would be a lot of help.  Thanks

REPLY [16 votes]: Let me explain two ways to derive the parametric equations:

Coordinate transformations:

Think that you are in an amusement park and you are represented by a stick which rotates with respect to 2 axes at the same time. With respect to the big pole in the center of the rotating machine, and with respect to a center attached to the basket holding you. The horizontal rotation is moving at an angular speed $\omega$ and the vertical plane rotation (your basket) set to a different velocity $\omega/2$. 
If we take snapshots and gather them together we
will observe something similar to what is drawn in Figure 1 below.

We sampled the circumference every 10 degrees. If instead of sampling
   the circumference we run the sticks along a continuous path, we would
   have a recreation of the “half” twist Mobius strip. We say “half”
   because the rotation of the sticks is that of 180 degrees. We will
   generalize the concept of a rotating segment for a concept of a
   general f (x, 0, z) path in the vertical plane and from it compute the
   Mobius strip, but many Mobius-like solids can be computed with this
   method, with several cross-section shapes and any multiple twists of
   $\pi$.
More generally:   We consider a curve in the vertical plane $(x,0,z)$
   given by an implicit equation   $f(x,y,z)=0$, or better, by
   parametric equations
\begin{eqnarray}
  x = x(s) \quad , \quad
  y = y(s) \quad \text{and} \quad
  z = z(s).
  \end{eqnarray}
and we want to apply the following transformations to this curve:

Rotatate the curve an angle $t/2$ in the vertical plane $(x,0,z)$.
Shift the origin along the $x$ axis by a distance $R > 0$.
Rotatate the curve an angle $t$ in the horizontal plane $(x,y,0)$.

The Figure below illustrates this:

Initially we  start with the parametric function $x'=x'(s),  y'=y'(s):=0$, and $z'=z'(s)$ as a loop around the origin
  (gray color). We filled the interior of the loop for visualization purposes
  only but we are interested exclusively to whatever happens to the boundary which
  is the parametric function above. If the interior of the curve is taken into account, we will find "solid" Mobius bodies instead.
Here are the steps in the figure:

Rotate the gray curve by an angle $t/2$ with respect to
  the $y$ axis and obtain the purple curve.
Translate the purple curve (which will be always in a vertical plane)
  $(x,0,z)$ by a distance $R$, along the $x$ axis  and obtain the green curve.
Rotate the green curve around the $z$ axis by an angle $t$
  and obtain the pink curve.

Mathematically we write the three steps as follows:

Rotate by an angle $t/2$ about the $y$ axis
  \begin{eqnarray}
  \left ( 
  \begin{array}{c}
    x \\
    y \\
    z
  \end{array}
  \right )
  \mapsto
  \left ( 
  \begin{array}{ccc}
    \cos t/2 & 0 & -\sin t/2  \\
    0 & 1 & 0 \\
    \sin t/2 & 0 & \cos t/2 
  \end{array}
  \right )
  \left ( 
  \begin{array}{c}
    x' \\
    y' \\
    z'
  \end{array}
  \right )
  \end{eqnarray}
Translate by  $(R,0,0)$
\begin{eqnarray}
  \left ( 
  \begin{array}{c}
    x \\
    y \\
    z
  \end{array}
  \right )
  \mapsto
  +
  \left ( 
  \begin{array}{c}
     R \\
    0 \\
    0 
  \end{array}
  \right )
  \end{eqnarray}
Rotate by an angle $t$ about the $z$ axis
\begin{eqnarray}
  \left ( 
  \begin{array}{c}
    x \\
    y \\
    z
  \end{array}
  \right )
  \mapsto
  \left ( 
  \begin{array}{ccc}
    \cos t &  -\sin t & 0  \\
    \sin t &  \cos t  & 0 \\
    0 & 0 & 1 
  \end{array}
  \right )
  \left ( 
  \begin{array}{c}
    x' \\
    y' \\
    z'
  \end{array}
  \right )
  \end{eqnarray}

The three steps applied together produce
  \begin{eqnarray}
  \left ( 
  \begin{array}{c}
x \\
y \\
z
  \end{array}
  \right )
  =
  \left ( 
  \begin{array}{ccc}
\cos t &  -\sin t & 0  \\
\sin t &  \cos t  & 0 \\
0 & 0 & 1 
  \end{array}
  \right )
  \left [ 
  \left (
  \begin{array}{c}
R \\
0 \\
0
  \end{array}
  \right )
  +
  \left ( 
  \begin{array}{ccc}
\cos t/2 & 0 & -\sin t/2  \\
0 & 1 & 0 \\
\sin t/2 & 0 & \cos t/2 
  \end{array}
  \right )
  \left ( 
  \begin{array}{c}
x' \\
y' \\
z'
  \end{array}
  \right )
\right ]
\end{eqnarray}
The expansion of this expression produces the following equations:
\begin{eqnarray}
  x &=& \cos t \left ( R - \sin \frac{t}{2} z' + \cos \frac{t}{2} x' \right ) -
  \sin t y'  \\
  y &=& \sin t \left ( R - \sin \frac{t}{2} z' + \cos \frac{t}{2} x' \right ) +
  \cos t y' \quad \quad (1)  \\
  z &=& \cos \frac{t}{2} z' + \sin \frac{t}{2} x' 
\end{eqnarray}
For the case of a M\"{o}bius strip we can start with a segment parametrized by
\begin{eqnarray}
  x' = s \\ 
  y' = 0 \\
  z' = 0
\end{eqnarray}
where $s \in [w/2,w/2]$ and $w/2 < R$. If $w/2 \ge R$ the surface will have
self-intersections.
In particular substituting this into equations (1)
we find
\begin{eqnarray}
  x &=& \cos t \left ( R  + \cos \frac{t}{2} s \right )  \\
  y &=& \sin t \left ( R  + \cos \frac{t}{2} s \right ) 
\quad \quad (1.2) \\
  z &=&  \sin \frac{t}{2} s 
\end{eqnarray}
which is the parametric equation of the Mobius strip.
It is interesting that:

If instead of the polar angle to be $t/2$ we choose $n t/2$ we can have as many "half" twists as we want by choosing $n=1,2, \cdots$. 
If we want a 1D manifold (a Mobius curve) we just fix $s$. So the edge
of the Mobius strip is given by the equation above for $s=w/2$ as a parametric equation only in $t$. Of course $s=0$ produces the circle of support (base).
We can generate 3D Manifolds by choosing an initial path other than a segment. For example $x'=a \cos s, \quad y'=0,  \quad z'=b \sin s$ (an ellipse). If $a=b$ we find a torus with circular cross-section. A twisted torus with $a=b$ and  with any $n=0,1,2,\cdots$ is a torus anyway since a circle is invariant under rotation. I would choose $n=0$ to get the simplest torus equation.In the case of a torus, if we let $R \to 0$, we move from a torus to a dullo and end up in an sphere at $R=0$. If $a >> b$ then we can see 3D Mobius bodies which resemble the Mobius strip.  If $a > b$ and $n=0$ we find a torus with elliptical cross-section.
If the polar angle $t/2$ is fixed to 0, and the generating curve a segment (as in the Mobius strip)  we get a flat ring horizontally laying on a flat surface. If $t/2$ is fixed to $\pi/2$ and the generating curve a segment (as in the Mobius strip) then we have a cylinder. If the polar angle $t/2$ is fixed to other value $\theta$,  we find a conical shape. This conical shape turns into a cone if $R \sin \theta = w/2$. After that (bigger $w$ values)  it crosses into an inverted cone.
2. Ruled Surfaces:

A different approach to this problem is by the use of
https://en.wikipedia.org/wiki/Ruled_surface}
Ruled Surfaces.
Look at the first figure above again (with the rotating sticks).
This suggest that the Mobius strip can be generated as a collection
of segments defined by the equation:
\begin{eqnarray}
\Sigma(s,t) = p(t) + s u(t).  \quad \quad (2) 
\end{eqnarray}
Here $u$ is a unit vector which points in the direction of the segment.
In the case of a Mobius band we have that the vector $u(t)$ is given
by the polar-spherical coordinates
\begin{equation}
u(t) = (\cos \frac{t}{2} \cos t, \cos \frac{t}{2} \sin t, \sin \frac{t}{2})
\end{equation}
and $p(t)$ is the point sitting at the origin of the segment (in this case
in the middle of it). The points $p(t)$ are sitting in a curve called
directrix curve  which in the case of the M\"{o}bius band is a circle
with radius $R$. For the case of the Mobius strip we have that 
\begin{equation}
  p(t) = (R \cos t, R \sin t, 0).
\end{equation}
By substitution of the origin and direction vectors into the surface
(2)  we arrive to the same parametric equations(1.2) for the 
Mobius band. The straight segments with directions $u(t)$ are called
 generators  of the surface. While some surfaces can have multiple
generators, the Mobius band has only one generator so it is called a single ruled surface.
The result is shown in the next Figure:

Note: I used https://www.sharelatex.com/learn/TikZ_package
to draw the three figures in this post. I desire to provide the tikz code for any or all of them to whoever wants it.