Hindging a Moebius Strip Kangaroo

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Permalink Reply by Daniel Piker on January 7, 2014 at 9:01am

Here is a definition which creates a developable looped strip mesh with any number of twists you choose.

At first this surface is doubly curved, but by setting regular edge lengths and relaxing (with a smoothing force to prevent crumpling), it can be made developable.

Attachments:

• moebius_developable.gh, 19 KB

Permalink Reply by JS on January 7, 2014 at 12:48pm

That's amazing Daniel! Thank you!

If you don't mind, could you explain a few things?

I tried to recreate your definition, but for some reason I get this gap. This is really frustrating because I even tried using the same numbers you did!

Then you used three spring components. Is this out of my league as a beginner to understand? I can follow the steps but I don't know why you are doing what you are doing.

You separated the mesh into two lists of lines, then found the average length and the springs are creating a force that is moving all the lines towards that average. So this is pushing the top and bottom of the ring towards the center?

The second spring I don't understand why it is necessary, but it must be important since when I remove it, the whole thing dies. It's also getting an average but of the short lines, which all kinda have the same size, so I'm lost as to why it is so much.

And the third spring is really confusing! You are creating diagonal lines, the I'm guessing the square rooting and adding is Pythagorean theory? But why do you need this spring at all? I can see it's also important, but I just don't know the rational behind it or why it is doing.

Thanks for the help Daniel. I can see I have a long way to go.

Attachments:

• moebius_developable.gh, 27 KB

Permalink Reply by Daniel Piker on January 7, 2014 at 3:42pm

To fix that gap I added the first item in the list of circles onto the end so that it closes the loop. When recreating it you probably just connected them in a different order. I should really have shown it like this to make it clearer:

Now to explain the springs -

The aim is to have the edge lengths correspond to a regular rectangular grid (so it could unroll to a flat straight strip).

The quads will be rectangles, but not necessarily squares, depending on the width of our strip and number of divisions. So we separate the edges into warp/weft directions and get the average for each to give us an average height/width for the rectangles.

With springs in only 2 directions, the rectangles can still distort into parallelograms, so that's why we need the diagonals (and as you correctly guess, they get their length using Pythagoras' theorem).

Permalink Reply by JS on January 8, 2014 at 11:53am

Thanks Daniel! I think I'm beginning to understand.