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Thanks.

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- Thread starter amcavoy
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Thanks.

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Hurkyl

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Certainly. Any function f from [0, 1]² → **R**³ 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.

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Thanks again.

- #4

Hurkyl

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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,

- #5

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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.

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.

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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.

Thanks for your help.

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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)

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Dr Avalanchez said:

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)

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apmcavoy said:

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

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