Understanding the Equation of a Möbius Strip

Question

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

Answer 1

Hmmm. You want an equation, but I have to not use "math jargon." And you want me to explain how to derive them, but presumably I'm not allowed to use words like "parametric." This does present a challenge.

I'm going to start with a circle in the $xy$-plane of 3-space. There are two ways to describe it: $$ x^2 + y^2 = 1 \text{ and } z = 0, $$ or $$ t \mapsto (\cos t, \sin t, 0) \text{, where $t$ ranges from $0$ to $2 \pi$}. $$

The first of these is called "the implicit form", because there's no way, just looking at it, to produce a point $(x, y, z)$ that satisfies both equations. (Yeah, it's clear that $z$ has to be $0$. And if you then try $x = 0$, you can guess that $y = 1$ and $y= -1$ work. But what if the $xy$ part of the equation had been sometime like $13x^2 - 11xy + 13 x + 2y - 3y^2 = -2$? Then you'd be in a pickle. So this kind of description of a shape is called "implicit" because it only lets you test whether a point's part of the shape or not, but doesn't explicitly produce any points.)

The second is called the "parametric form", because there's a "parameter" ($t$) that you can vary to generate points on the shape. As $t$ is varied from $0$ to $2 \pi$, you generate every point of the curve. In many situations, this kind of description is preferable, although there are also cases where the implicit description is better. We use both in mathematics. The parametric form has one disadvantage: sometimes two different parameter values (like $t = 0$ and $t = 2\pi$) correspond to the same location on the shape. There's a reason for that: the interval $[0, 2\pi]$ is a fundamentally different shape from a circle. Because of that, there's no "nice" way to send points of the interval to points of the circle and vice-versa and have the mapping be a one-to-one correspondence. (That's hard to prove thoroughly, but it's true.)

So what about surfaces? Well, for those, we need two parameters, like "latitude" and "longitude", to describe each point of the surface. Once again, we'll have the "colliding parameters" problem. On the earth, for instance, longitude 180W and 180E both correspond to points of the international dateline, and when you look at longitude-latitude pairs where the latitude is 90N, all possible longitudes correspond to the same point -- the North Pole.

The first thing I'm going to do is to describe a cylinder using two parameters. Once again, $t$ will tell us where we are in the "around the circle" direction, but I'll use a new parameter, $s$, to say how high on the cylinder we are. (The cylinder's aligned like a can of beans sitting on a table in this example): $$ (t, s) \mapsto (\cos t, \sin t, s) \text{, where $0 \le t \le 2\pi$ and $-1 \le s \le 1$}. $$ I've cut off $s$ at $-1$ and $1$ to make a cylinder of height 2.

Roughly speaking, at each point of the unit circle, by varying $s$ I can move up and down in the $z$-direction.

For a Mobius strip, you also want to move "perpendicular to the core circle," but you don't want to always move up and down; you want to move in a "tilted" direction. So I'm going to rewrite what I wrote above for the cylinder in a new form:

$$ (t, s) \mapsto (\cos t, \sin t, 0) + s (0, 0, 1) \text{, where $0 \le t \le 2\pi$ and $-1 \le s \le 1$}. $$ In that form, you can see that we're starting at a point of the circle, and adding to it a displacement in the direction $(0, 0, 1)$, with the amount of the displacement goverened by $s$.

To make a Mobius band, we need to change that displacement direction to one that rotates as we move around the circle. That is, we want to write

$$ (t, s) \mapsto (\cos t, \sin t, 0) + s v(t) \text{, where $0 \le t \le 2\pi$ and $-1 \le s \le 1$}. $$

where $v(t)$ is a direction that changes when we vary $t$. At $t = 0$, we want it to point straight up. By the time we reach $t = \pi$, we want it to point in the $(-1, 0, 0)$ direction. And by the time we reach $t = 2\pi$, we want it to point straight down.

To build $v(t)$, I'm going to combine the straight-up direction, $(0, 0, 1)$ with the "pointing outward in the $xy$-plane direction", $(\cos t, \sin t, 0)$ in a way that varies as a function of $t$. Here goes:

$$ v(t) = \cos(t/2) (0, 0, 1) + \sin(t/2) (\cos t, \sin t, 0). $$

The reason for the $t/2$ is that as $t$ ranges from $0$ to $2\pi$, I wanted the angle that the ray was pointing to rotate only half a turn. Combining all that, the final parametric description is

\begin{align} (t, s) &\mapsto (\cos t, \sin t, 0) + s (\cos(t/2) (0, 0, 1) + \sin(t/2) (\cos t, \sin t, 0))\\ &= (\cos t, \sin t, 0) + (0, 0, s\cos(t/2) ) + (s\sin(t/2)\cos t,s \sin(t/2)\sin t, 0)\\ &= (\cos t + s\sin(t/2)\cos t, \sin t + s \sin(t/2)\sin t, s\cos(t/2) ). \end{align}

I hope that's of some help.

By the way, I entirely agree with Daniel Rust that the "strip with identified edges" description of the band is more useful in almost every context. But sometimes it's nice to be able to write things down explicitly, too.