Folding Paper

I'm bending a quadrilateral piece of paper along one diagonal. I've modeled a rigid version, but I want to make a surface that most closely approximates what a piece of paper would be shaped like, using the same 4 edges, but without creasing the diagonal fold. When folding it by hand, it just appears as a strip of paper that is twisting.

So I'm looking for advice on how to make the paper surface. (holding area of paper the same)

See images and definition file below, if you have any ideas. Thanks.

Here are videos of the piece that is made up of these surfaces:

http://www.grasshopper3d.com/video/hyperbolic-paraboloid-origami

http://www.grasshopper3d.com/video/hyperbolic-paraboloid-origami-1

Attachments:

CurvedPaper.ghx, 154 KB
CurvedPaper.3dm, 52 KB

Replies to This Discussion

Permalink Reply by Tomohiro Tachi on March 4, 2010 at 7:41pm

You might be interested in the results of our paper "(Non)existence of Pleated Folds" presented at JCCGG 2009, which proves non-existence of the non-triangulated hypar model (and the existence of triangulated models).

One of the theorems we have proved is that a piece of developable surface bounded by straight crease lines is polygonal (piecewise linear) in a half-folded state.
A smooth surface between ABCD must always have negative Gaussian curvature like Srf4Pt.

Permalink Reply by Chris Wilkins on March 4, 2010 at 8:04pm

I'm definitely interested in your paper. Do you have a link to it ? (before the Googling commences)

So is Srf4Pt a good approximation of the surface, or is it identical, or just similar?

negative Gaussian curvature...hmm... guess I have some research to do...

Permalink Reply by Tomohiro Tachi on March 4, 2010 at 8:31pm

Previous version of the longer paper is posted in arxiv.
http://erikdemaine.org/papers/Hypar_JCCGG2009/

Developable surface, i.e., ideal paper, (a surface not allowed to stretch) has zero Gaussian curvature and cannot fit a twisted quadrilateral smoothly.
So your smooth pleated hypar is somehow "wrong," or the surface is slightly stretched. I think Srf4Pt is a good approximation in the sense that it looks nice.

Permalink Reply by Chris Wilkins on March 4, 2010 at 9:03pm

Thanks for the link.

The end result for next week's project is that I want to show the curvature of the surfaces using a gradient.

Permalink Reply by Chris Wilkins on March 14, 2010 at 2:39pm

I've just been reading through your paper (the long version). Very helpful towards understanding this topic. Thanks.

On page 13, it says: "Interestingly, the diagonal cross-sections of these structures seem to approach parabolic in the limit." You said that the error was extremely small when the number of rings was 100. After your paper was written, did anyone figure out whether it just SEEMS to approach parabolic, or does it ACTUALLY approach parabolic? (if number of rings approaches infinity)

Is it possible to evaluate this shape approaching infinity? Or does the equation for the position of the vertices just keep growing the more rings you add?

My teacher wants me to take my Grasshopper definition and code it in VB so that we could vary the number of rings and see how the shape approached parabolic. (So far I just keep copy and pasting and rewiring the same tangle of components to make more rings). I found your Mathematica construction algorithm of the triangulated hyperbolic paraboloid at the end of the paper. I can't read Mathematica, but I'm wondering if I try and study through it, whether it might help me make a Grasshopper version. Have you guys done any further work on this since your paper was written? Thanks!

Permalink Reply by Mårten Nettelbladt on March 5, 2010 at 1:07am

I would make a large paper model and study it very closely.
Do the sides of the paper remain straight or do they curve?
Can you find any straight "lines" in the the curved paper shape?

It is quite possible to connect those 4 points (A,B,C,D) with a developable surface if you don't have to follow the straight edge lines (AB,BC,CD,DA).

Can you post a photo of a (fairly large) paper model that is showing the desired "twist"?
It just needs to be a sheet of paper that you hold in your hand.

Permalink Reply by Chris Wilkins on March 5, 2010 at 9:20am

Here are photos:

1. The whole piece:

2. Modeled from resin on 3D printer:

3. Here is folded and curved versions next to each other:

4. And another angle showing the curve and the fold:

Permalink Reply by Mårten Nettelbladt on March 5, 2010 at 1:23pm

Hi Chris, I tried to replicate your model.

(1) First test is with loose edges.

(2) The edges start to curve quite a lot to deal with the twist:
(Cone sections are seen, with centre points in A, B and C)

(3)

(4)Then I made the edges rigid (like in your model):

(5) The paper deals with this in two ways, one is crumpling:

(6) The other way is curved edges:

(7) The curved edges allow for a developable shape that consits of two cone sections (joined by the green line, which is also the location of your extra fold.) Centre points are A and C.

I guess crumpling is tricky to do in Grasshopper (perhaps something for Kangaroo?),
so I would use the two cone sections to create an accurate model in GH.

Or, actually, stick with the extra fold! (A folded surface is also developable, and you've already made lots of other folds).

Good luck!

Permalink Reply by Chris Wilkins on March 5, 2010 at 5:58pm

Mårten, Thanks very much for the detailed explanations. That illuminates several things I hadn't considered yet. I'm very new to this folding stuff. I didn't even intend to get into it, but it started as a diversion and turned out to be pretty fun. The rigid stuff seems somewhat straightforward, but the curved stuff is a bit beyond me at the moment. I knew that the crumpling was happening if you bend things too far, but I wasn't even going to attempt that. I'm also going to assume very stiff edges, and not let them bend at all. I'll post whatever further development I get to with this piece. Thanks again.

RSS