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Folds in a sheet of paper / metal / cloth

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I am an artist, drawing and working with 3d. Since long time, I am missing a method to form realistic folds and wrinkles into virtual sheets of paper, metal or cloth. I found some attempts within 3d-programs, but I never saw something that was not very limited or looked like rubber. Because it is so easy at reality, but so difficult at the computer, I think there is just little mathematical base.First of all: Could anybody tell me, what field of mathematics this is?




POSTED BY: Dietmar Klein
Answer
1 year ago
Dear Dietmar,

This is basically origami. It is all about crease-patterns, obeying several rules/constraints. And that can be solved by circle packing.

See especially this TED video:

Robert Lang: The math and magic of origami

enjoy emoticon
POSTED BY: Sander Huisman
Answer
1 year ago
I heard about this guy (and tried to contact him many years ago.) Yes, it is nearby. Maybe this is it,- but I am not sure: Think of it not at a way which leads to the optimum but as an accident which goes its physical way. I am longing for the process. What happens on the way from the flat sheet to 180°-fold? How tide can you go? What about the stiffnes of the sheet and the tendency to smooth all edges? Where is the point of break, from which you need more sharp angles? (And, by the way, I would not be able to transform his program into 3d-program.)
Thank you so far.
POSTED BY: Dietmar Klein
Answer
1 year ago
MIT posts their course on Geometric Folding Algorithms for free here:

Geometric Folding Algorithms: Linkages, Origami, Polyhedra

I watched one a while ago and thought it was very interesting. I had no idea the mathematics of paper folding was so well-developed!
POSTED BY: Michael Hale
Answer
1 year ago
This is an interesting question, and Sander and Michael started putting out ideas for resources. I thought it is a good idea for a start and I would note a couple of related things.

Robert Lang
==========


First of all Robert Lang, the presenter in the TED video linked by Sander, is an avid Mathematica user. For example quoting Rober Lang from McGill's "Origami pteranodon" :
I love Mathematica. TreeMaker (his own origami software) is a surgeon’s scalpel—a very narrow tool that does what it does incredibly well, but nothing else. Mathematica is the world’s largest Swiss Army knife. You can do anything in it: analyze, compute, simulate, visualize. And almost any project that involves something I’ve never done before probably involves Mathematicaat some point. With Mathematica, I can import things. I can manipulate them, I can do visualization. I can solve equations, I can do algebraic manipulations and 3D renderings. I can plot surface graphs. It’s just mind-boggling.

Just in case here is another link. Robert Lang is also an author at Wolfram Demonstrations where the source code notebooks can be freely downloaded:



You can find moch more at Demonstrations Project - just search, for example: folding or origami or paper. According to his recent interview The Art of Origami we can again quote Robert Lang:
I also have a fairly large Mathematica package, called Tessellatica, which contains tools for designing, constructing, and visualising origami tessellations of broad variety and runs within the Mathematica environment, but it’s not publicly released yet.
- something to keep in mind.

Yves Klett
=========


Another great Mathematica user Yves Klett has had presented at Wolfram Technology conference about folding - see for example: Aerospace Origami & Fractal Folding and the video An Engineer's Box of Chocolates.

Research
=========


The following research paper

(Non)existence of Pleated Folds: How Paper Folds Between Creases
Erik D. Demaine, Martin L. Demaine, Vi Hart, Gregory N. Price, Tomohiro Tachi

has extensive discussion on creases with given Mathematica code. If you check the authors affiliations and do a little web search on them, you will find them to be very interesting creative people. Here is a snapshot from the paper:



I hope more discussion will follow with more precise approaches.
POSTED BY: Vitaliy Kaurov
Answer
1 year ago
Thanks for supporting. I did not yet study all, and I will spend some time with this. Up to now I have the feeling, that origami is a part, but differs. Maybe its a matter of dimension (3 instead of 2? 4 instead of 3?;-) or random or physics. This is a spontaneous try to focuse an example:



PS.: Of course I saw the "stent",- this drawing is not a 100% serious. But up to now I lack words to descibe the difference.
POSTED BY: Dietmar Klein
Answer
1 year ago
Hi Dietmar

A while ago (Fiac 2007-8 maybe), in a famous garden in Paris, I have seen what seems to be huge sheet of wrinkled metal, exactly like a paper sheets, the one you throw to the trash can. The artist who made that is Wang Du  ( some samples here ). I am still wondering of how he succeed to do it, is it reproduction at scale or is it made using a mechanical process ? It was metal and seems to have been made by a geant hand...

The right tool to simulate that kind of shaping is finite element, with some special hypothesis (thin material). Those kind of tools are used in the automotive industry for crash test simulation.  But even if you could access such a tool (witch usually run on a supercomputer), for artistic design, I am not sure there is a way to design any shape in an other way than experiment on scaled models. And it is maybe not even reproducable, even with a folding robot that apply repeatable constraints because of instable effects. This let randomness play a good role for art creativity, anyway.

Nicolas
POSTED BY: Nicolas Venuti
Answer
1 year ago
Here is a link to what Nicolas is talking about: FEM - Finite Element Method. In addition I also would like to mention something called curved folding - which is basically something we already touched upon earlier. Still take a look at this beautiful site:

CURVED FOLDING
Martin Kilian, Simon Floery, Zhonggui Chen, Niloy J. Mitra, Alla Sheffer, Helmut Pottmann
ACM SIGGRAPH 2008

And also take a look at the attached PDF papers at the bottom of the site. Some images look like almost what you need, and give some schematics.





POSTED BY: Vitaliy Kaurov
Answer
1 year ago
To Nicolas: There are so many different things he does,- I am nearly sure that Wang Du does small modells and gives them to a company. There they scan the model, make a big reproduction and do the finish by touching up, polishing and so on. Demian Hirst works similar, or 1 generation earlier: Andy Warhol or, roughly 20 generations earlier, Rembrandt.
But I am not longing for art,- I want to model with 3d.
FEM could it be. I understand nothing, but: Isnt there any chance to break this down for just visual purposes? By shortening the lenght of computing until the xx. step or details less than xx whatevers? By adding a bit of fuzzy? Or by limiting on 5 to 10 standard situations?
POSTED BY: Dietmar Klein
Answer
1 year ago
Hi Dietmar

There are many issue with FEM, it is slow but with some simplifications i thing it could be real time on modern PC. FEM software are not designed for interactivity like 3D cad software, also the interaction with the material must be done in a mathematical way by defining deformation or effort you apply on the material. It is far to be as easy to use as 3D software. With a FEM software you could deform the material in a realistic way, and go to a final result depending of the path of your physical interactions, but you cannot design directly the final object assuming constraints of the material are respected.

What seems possible is to use your prefered CAD software and design your shape using only surface primitives,
assuming in you mind that it is a perfect sheet at the origin, which obviously requires experience. 
Then, create a polygon approximation of the shape (the CAD software is maybe able to do it for you)
Feed the result in some Mathematica code to compute the constraint that would appear by flattening your design back to the original sheet form.
For example, use a color map, to display on the polygon where the tensile effort are the biggest positive (red) and lowest negative (blue)
Then map back the colored polygon to your original shape, by reimporting in the CAD software.
Then use a design heuristic :
- where the tensile force is to big you have to deform the shape to lower it (give locally more surface to the shape by changing the surounding)
- where the tensile force is to small you have to add some wrinkled (remove locally some surface)
This is just an idea, but if you could add this feature in real time superimposed to your software,
you could have a desgin assistant to follow the perfect sheet deformation constraint...

You can also try to use a physics engine created for games, they can approximate clothes or flat material by a chain, and are realistic.

Nicolas
POSTED BY: Nicolas Venuti
Answer
1 year ago
Thanks for your interests, Nicolas,
I do not get all , because I am not at home at english nore at mathematical terminology. But I think, something like this would work. Please wait some days. I would like to think about the very needs, do another sketches and show you again.

Dietmar
POSTED BY: Dietmar Klein
Answer
1 year ago
I thought of something like an universal fold, which can be combined to any folds and wrinkles. But I cant fix it up to now, so I close this discussion. Thanks for your ideas and suggestions. Maybe another time!
Dietmar
POSTED BY: Dietmar Klein
Answer
1 year ago
A video from Siggraph, definitely have a look:

Folding and Crumpling Adaptive Sheets, SIGGRAPH 2013
POSTED BY: Sander Huisman
Answer
1 year ago
Sander, this is a great find! I think it is exactly what is needed. Here is exact citation with abstract an image that can be found on the authors Berkeley website ( there is also full paper and source code download):
Rahul Narain, Tobias Pfaff, and James F. O'Brien. "Folding and Crumpling Adaptive Sheets". ACM Transactions on Graphics, 32(4):51:1–8, July 2013. Proceedings of ACM SIGGRAPH 2013, Anaheim.

We present a technique for simulating plastic deformation in sheets of thin materials, such as crumpled paper, dented metal, and wrinkled cloth. Our simulation uses a framework of adaptive mesh refinement to dynamically align mesh edges with folds and creases. This framework allows efficient modeling of sharp features and avoids bend locking that would be otherwise caused by stiff in-plane behavior. By using an explicit plastic embedding space we prevent remeshing from causing shape diffusion. We include several examples demonstrating that the resulting method realistically simulates the behavior of thin sheets as they fold and crumple.

POSTED BY: Vitaliy Kaurov
Answer
1 year ago