Parametrization of a Moebius Strip

[amcavoy]

I was wondering about the different methods by which one could "parametrize" a Moebius Strip. I asked someone about this a while ago, and they said that since the center of a Moebius Strip (z=0) is a circle, you can begin with the parametric equations for that and draw vectors out to other points on the strip. Is there another way to do this?

Thanks.

 

[Hurkyl]

Certainly. Any function f from [0, 1]² → R³ with f(0, x) = f(1, 1-x) would suffice (although many would be self-intersecting). I can't think of another easy way to do it, though.

 

[amcavoy]

Alright. I never got anywhere trying to do it with vectors (although I know I would have to introduce a new variable). How would you begin to do this?

Thanks again.

 

[Hurkyl]

No matter how you do it, your parametrization has to have two variables...


Start off by trying to figure out, in words, how to select a point on the Möbius strip by using the pair of numbers (u, v). Once you can work out these details, then it should be straightforward to write it down.

 

[Antiphon]

It's not as simple as it looks.

If you just try to parameterize the edges it will work out. But when you parameterize
the surface, you will find that there are two different parameter sets
corresponding to each point on the surface. In other words, the
obvious parameterization is not single-valued.

Try it with a pencil and paper and you'll see what happens.

 

[amcavoy]

Yes, looking at the equations themselves it all makes sense. I just didn't know whether to introduce a new parameter as an angle, or a portion of the half-width. Can a Klein Bottle be done in a similar way?

Thanks for your help.

 

[Dr Avalanchez]

Sorry bout being a bit late, but this is how you could see the creation of a mobius strip:

Let R>1:
Rotate the line [R,0,u] (-1<=u<=1) in the XZ-plane over an angle of v/2 around the center of the line.
Rotate the line over an angle v around the Z-axis.

The transformation thus consists of a translation, rotation round Y, inverse translation, rotation round Z.

(I have a maple file with all the steps worked out in full detail if you're interested)

 

[amcavoy]

Sorry bout being a bit late, but this is how you could see the creation of a mobius strip:

Let R>1:
Rotate the line [R,0,u] (-1<=u<=1) in the XZ-plane over an angle of v/2 around the center of the line.
Rotate the line over an angle v around the Z-axis.

The transformation thus consists of a translation, rotation round Y, inverse translation, rotation round Z.

(I have a maple file with all the steps worked out in full detail if you're interested)

I am interested. Please send it (or upload it here) if possible.

The way I looked at it was to take the radius of the circle on the plane z=0. Let this be R. Looking at the strip, it seems clear that the angle the strip makes with the xy plane is t/2, where t is the parameter of the circle of radius R. Now introduce a new variable m so that m∈[-n,n] where n is the half-width of the strip. The equations seemed to follow:

x=[R+n*cos(t/2)]*cos(t)
y=[R+n*cos(t/2)]*sin(t)
z=n*sin(t/2)

 

[Dr Avalanchez]

Rename the file to .mw

 

[amcavoy]

Great thanks a lot. By the way, is there a similar way to do this for a Klein Bottle? It seems a bit more elusive.

 

[Dr Avalanchez]

Great thanks a lot. By the way, is there a similar way to do this for a Klein Bottle? It seems a bit more elusive.

Probably, but I'll have to think about it. (don't hold your breath, I'm in the middle of exams)