Folded plane subdivided into equilateral triangles

Replies to This Discussion

Permalink Reply by C.Render on February 12, 2014 at 6:38am

Here is the image

Attachments:

• IMAG2553.jpg, 102 KB

Permalink Reply by Benjamin Felbrich on March 3, 2014 at 3:45am

Hey C.Render,

how exact do you want your solution to be??
You could use Kangaroo: Make a mesh out of your surface. Then, with Kangaroo, you can turn every Mesh Edge into a spring and tell the engine to pull it towards a certain length. You might need to play around with it a little bit and also change the topology of the Mesh.

Permalink Reply by C.Render on March 3, 2014 at 6:54am

With Kangaroo I doesn't come along. I tried it so far with LunchBox. But as you can see at the attached picture, there some "mistakes":

 1. on the left and on the right side are triangles I don't want

 2. the triangles just "seems" like they are equilateral, because I just move the number slider to a position, that makes them look like, but I want really equilateral triangles

3. how can I move the ponits now (and so form my objekt) without changing the size of the triangles?

Permalink Reply by Hannes Löschke on March 3, 2014 at 8:36am

Six eqilateral triangles connected to form a hexagon will always be planar. So from a mathematical point of view you can have either

 - a free form surface

or

 - a surface panelled flush with eqilateral triangles

The form created in your picture results from stretching and tearing the paper at the folds and probably some bending of the triangle edges. So either the "triangles" aren't planar or they don't connect flush at the edges. That's why Benjamin asked you for the desired precision.

btw.: I cannot see any files attached to your last post.

Permalink Reply by C.Render on March 3, 2014 at 9:06am

Oh sorry, i've forgotten to attached it, here it is:Maybe you're right. It was just an abstract modell. But it should be very nearly exact and the triangles should be planar also. I know, that this must work, but until now, I haven't found out how it works. In his video Benjamin shows that there are a solution

http://www.grasshopper3d.com/video/standardized-surface-triangulati...

Permalink Reply by C.Render on March 3, 2014 at 9:21am

Here are some examples with equilaterale triangles:

Permalink Reply by Benjamin Felbrich on March 3, 2014 at 9:49am

Hi Chris,

in the video you referred to I am using three diifferent edge lengths. They are not necessarily equilateral. There's a set of 11 availabe triangle configurations of which some are equilateral. The algorithm choses the best fitting one in each step. The smoothness you can obtain increases exponentially with the amount of available edge lengthes.
So when you want to use only one available edge length (to get equilateral triangles) the result might always be bumpy and you won't get super smooth results, unless you accept some inaccuracies (which sucks, i know).

I wrote about that in my diploma thesis. Maybe it helps you a bit, here's the part about the geometry:

http://we.tl/pDs78KoOwo

Permalink Reply by C.Render on March 3, 2014 at 10:24am

Hi Benjamin,

first of all thank you for your attachement! I hope it will help me a littel bit.

Realting to my model: It is not necessary to be a smooth objekt. I rather appriciate some irreguality, because that would appear my model more realistic, more like a "folded piece of paper"

Permalink Reply by Benjamin Felbrich on March 3, 2014 at 11:02am

Hmm if the smoothness is not important to you, i would actually recommend you to try the Kangaroo approach.

turn your surface into a mesh and then tell kangaroo to equalize the meshes edge lengths. At the same time, you'll need to pull the mesh vertices towards your initial surface. its about balancing these forces. 
if you're new to GH it might be a bit tricky at first, but Kangaroo is super useful anyways. So it's totally worth to give it a try

i attached something that could work.

Attachments:

• equilateral_triangles.gh, 15 KB

Permalink Reply by C.Render on March 3, 2014 at 12:28pm

Thank you for the script, but actually, what should i set as a surface? Should I first try to model some smothe plane?

If I set the yellow surface as a surface then an error appears:Please excuse my nesience, but I'm a very beginner at GH, but I want to learn it as soon as possible and I do my very best! So thank you so far for all your help!

Permalink Reply by C.Render on March 4, 2014 at 4:31am

Still doesn't work...Attached you find my model and the GH file

Attachments:

• Papiermodell.3dm, 526 KB
• equaliteral_triangle.gh, 16 KB

Permalink Reply by Daniel Piker on March 3, 2014 at 10:17am

It is not true that 6 equilateral triangles around a vertex have to be coplanar.

Here's a couple of simple counter-examples:

While it is true that free-form surfaces can't be tiled smoothly with equilateral triangles, crumpled versions like the one shown are possible, even with exactly preserved edge lengths and angles.

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