Straight line on a mobius strip

Question

I have a mobius strip that is twisted by 540° degrees (not only 180° as the usual mobius strip).
Also it is one that can't be constructed with a paper-strip.

I created it with this OpenSCAD code:

for(angle = [0:0.3:360])
    rotate([0, 0, angle])
        translate([80, 0, 0]
            rotate([0, -angle/2*3, 0])
                cube([60, 0.5, 1], center=true);

The result looks like this: Mobius Strip with 540° twist
So far so good.

Now I would like to wrap an LED-Strip around it.
But I'm just not able to figure out the mathematics.
I think maybe Geodesic could be the way to solve this.
But I also read that these are usually second order differential equations to solve which is way above my skills.

I'm looking for something like a formula or an algorithm of the path of the LED-Strip.

I tried to apply the LED-Strip by hand.
It worked out quite well but is just not perfect: First Try

Any help or hints are welcome. Many Thanks.

Answer 1

The Möbius strip in the CAD model and photograph are formed by revolving a line segment at constant speed in a longitudinal plane. When the segment revolves through three half-turns, this strip may be described parametrically as $$ x(\rho, \phi) = ((R + \rho\cos\tfrac{3}{2}\phi)\cos\phi, (R + \rho\cos\tfrac{3}{2}\phi)\sin\phi, \rho\sin\tfrac{3}{2}\phi),\qquad -r \leq \rho \leq r,\ 0 \leq \phi \leq 2\pi. $$ It's true that this strip cannot be made exactly out of paper. The issue is not that the surface contains a plane curve (all Möbius strips contain plane curves, where the strip is cut by a plane), but that is has non-zero Gaussian curvature. Non-zero Gaussian curvature is an issue with _every  familiar model, however, including the photograph of the strip made from actual paper, see note below.

If the LED strip is not sitting nicely on the surface despite the surface being smooth in the physical sense, that's likely due to the surface's non-zero Gaussian curvature. Qualitatively, an untwisted LED strip has the same length at each location across its width, but even a narrow band of surface has varying length at different locations across its width. The effect can be reduced by increasing the radius of the strip's central circle, or by using a different parametrization, but I'm not sure that suits the model-building purposes, and I don't know of a good parametrization to suggest offhand.

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Technical note: In Euclidean three-space, there exists no Möbius strip that is real-analytic and developable, i.e., with Gaussian curvature zero. (I first learned this as a student from Morris Hirsch.) This fact is possibly under-appreciated; it certainly is not quoted widely. In fact, a quick search yields a number of web sites that appear to claim otherwise.

Loosely, this means any surface we can write down with formulas (such as the parametrization above, which is real-analytic) does not give a mathematically-flat surface.

There do exist flat (and real-analytic) abstract Möbius strips. The claim about is about images of real-analytic parametrizations in Euclidean three-space.

There do exist smooth flat Möbius strips in Euclidean three-space, but the only examples I know of are not band-like. (They all contain planar regions where "the direction of bending changes".)