On Sat, Oct 3, 2009 at 1:34 AM, xxx xxxxx <xxxx@gmail.com> wrote:
> Hi Kyle,
>
> Is there a way to create a point cloud that restricts the location of points
> to a specified distance from eachother? For instance, If you folded up a
> piece of bristol with a 1″ x 1″ dia-grid on it the length between points
> (represented by the score lines) has to remain constant because of the
> nature of the paper. This is not to say that certain folds won’t be able to
> bring points closer together, but there is always a certain distance that
> some have to keep. I could elaborate more if you’d like. Do you know of a
> rhino script that might be able to help me with this?
>
> Thanks a bunch man, let me know what I can do to help out if you have time
> to think about this,
>
> -Tim
>

Hi Tim,
Apologies for not getting back with you sooner, and without better news… this problem is more difficult than you might imagine.  In fact, you are at least screwed in three ways I can think of.

The way you frame your question, in terms of restricting the movement of points, can certainly be done.  Think of building any parametric or relational model – you could start at one corner of the diagrid and work your way along, defining each successive point as lying on a circle (2d) or sphere (3d) centered on the previous point.  But what next?  I assume you’ll want to grab on to any of the points and throw them around – but the way in which you’ve constructed them is such that the position of each point relies on the position of the one you build previously.  Point 10 can move point 11, but not point 9.  What you’ve got is a pretty strict hierarchy, such that you would you would only be allowed to grab that first one and shake it around.  This will be true of any parametric system which doesn’t employ a “simultaneous solver”, and as far as I know only Catia and Solidworks have integrated such a thing into their 2d sketching environments, and don’t allow this kind of relationship in 3d.

To make matters worse, even if you were okay with just grabbing the first point there in the corner and throwing it around, you’re still screwed because I would imagine you would want this movement to not only effect the points surrounding this “master point”, but all the points in the array as well… the way that paper would behave.  This, too is possible, but would require some kind of energy distribution system, such as the runge-kutta method (see link below).

The third way you’re screwed in reproducing the behavior of your paper model has to do with the physics and physical qualities of the paper itself.  I’m sure your paper model is awesome.  It can expand and contract in all sorts of awesome ways.  Ask yourself – would it behave differently if it were made of vinyl rather than bristol?  What about rubber?  What if it were huge and made of sheet steel and hinges?  The behavior would be different, of course, with these different materials at these different scales.  The reason why is that the paper bends with a specific degree of elasticity and resistance along each fold line, and with a certain degree of flex and stretch along each panel.  It *could* be that given the geometry of your fold pattern that you would be unable to fold the paper *at all* if it were made of something completely rigid… like the sheet steel and hinges begin to approximate at large scales.  So, I would guess that many of the qualities that make your paper model so awesome simply won’t translate into the completely rigid geometric world of simple grasshopper or rhino scripts.  You would need something that takes the physics of elasticity and bending into account.

Far sharper minds than mine have looked into this problem – one of whom I’ve had the pleasure of working with, Erik Demaine at MIT.  His specialty is in the mathematics of origami – and he’s worked out alot of cool stuff through a really deep understanding of tiny bits of folding paper.  Below, I’ve included some links that I’ve found helpful while researching this topic.

solvers:
http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

software:
http://www.langorigami.com/science/4osme/abstracts/048%20Tachi.pdf
http://www.tsg.ne.jp/TT/software/index.html
http://mitani.cs.tsukuba.ac.jp/pukiwiki-oripa/index.php?ORIPA%3B%20Origami%20Pattern%20Editor
http://www.papermosaics.co.uk/software.html
http://www.langorigami.com/science/treemaker/treemaker5.php4

erik demaine:
http://www.nytimes.com/2005/02/15/science/15origami.html
http://erikdemaine.org/

origami:
http://design.origami.free.fr/Diagrams/cp.htm
http://www.richardsweeney.co.uk/