Schramm circle pattern

Comment

Comment by Dedackelzucht on August 16, 2012 at 7:40am

Hey,

2 out of 3 orientations of the frames (corners where 4 circles touch) are fixed. If you see it as a strict rectengular grid the x and y direction have alternate positive and negative the same angle. this angel you can change it has a impact on the radius of the edge-arcs of each patch.

Right now I have too much to do . I cannot spent some more time now in it... :-(

Thanks again

DeDackel

Comment by Daniel Piker on August 15, 2012 at 6:00pm

DeDackel,

This could be something really interesting to try on a circular quad mesh.

(there are already some examples up of how equalization forces in Kangaroo can be used to generate such a mesh, where the vertices of each quad lie on a circle. I'll try and post some more examples soon though).

Then you could fit a cyclide patch like you show below to each quad/circle of the mesh.

For a given circular mesh, you still have a choice of how you orient the corner frames - though once you fix one, that determines all the others through mirroring operations.

I think it should also be possible to use Kangaroo to generate and transform cyclidic nets directly, based on the 'cyclidic cube' construction in that first paper.

Lots of fun things to try!

Comment by Dedackelzucht on August 15, 2012 at 6:22am

Hey Daniel,

yes, thats exactly how I did it. Funny! Like in the script explained I started with the "DUPIN CYCLIDES" and then connected them together to "CYCLIDIC NETS" based on four circles.

Where do you have all so cool pdfs from? :-)

Best Regards

DeDackel

Comment by Daniel Piker on August 15, 2012 at 5:41am

Hi Dedackelzucht,

Very interesting. So are you generating cyclide patches, like described in this paper by Bobenko and Huhnen-Venedey

http://arxiv.org/pdf/1101.5955.pdf ?

(see also http://page.math.tu-berlin.de/~huhnen/talks/12_ANU.pdf

and http://www.geometrie.tugraz.at/wallner/cas.pdf)

Comment by Dedackelzucht on August 15, 2012 at 2:31am

Hey Daniel,

That is very nice. I have been working on a defenition for this as well and did not get good solutions. In 2D its ok, but 3D is quiet complicated. The special thing about it for me is, that you can make groups of knots with 4 circles (all tangent in one point). and then use arcs to genrate shapes for planar quads.

Have you used Schramms Mathematical functions for the grid? I tried to get a soluttion by the tangencies of many circles controled by 3 pts.

http://www.ics.uci.edu/~eppstein/junkyard/tangencies/octahedron.html

nice work!

Best Regards

DeDackel