Rheotomic surfaces and flowline generation tool

Around 3 years ago I wrote an essay on my blog about what I called rheotomic surfaces - a type of surface I had developed related to fluid flow and electrostatics, and a technique for their generation using complex numbers.

Since then I have received a lot of questions from people interested in the details of exactly how these surfaces and their associated curvilinear orthogonal grids were generated.

Now I've packaged it up into a Grasshopper object with an easy interface, and am releasing it publicly so anyone can experiment with this tool.

(See this video for an example of it in action)

When the idea of using the streamlines of a flow to generate a surface first occurred to me, I thought the way to go about this would be to integrate a 2d vector field from various seed points and then move these lines vertically and loft between them - but after a lot of head scratching and experimentation, I was amazed to discover that it is actually possible to skip that step altogether.

In this technique, the surface is generated first, by moving the points of a mesh vertically from the complex plane according to the scalar values of their real and imaginary components, to generate 2 separate meshes. One of these meshes gives the rheotomic surfaces described in my essay, with helicoid shaped regions near the sources and sinks, and its contours are the streamlines of the flow (hence the name). The other mesh has sharp funnel shaped regions, and its contours give the equipotentials of the flow, orthogonal to the streamlines.

One of the advantages this technique has over vector field integration methods is that there is no problem of choosing seed points for streamline placement, and nice even spacing happens automatically. We also avoid the difficulties with cumulative error common to such methods.

By multiplying by other complex factors it is also possible to generate lines at specific angles to the streamline/equipotential directions and create various grid types.

Also because of the mesh contouring technique, these are actual vector curves being created, not just pixel based mappings.

Because the complex logarithm function is multivalued, dealing with the mesh in a way that avoids a sudden jump at the branch cuts does require a bit of special treatment, and it is not quite a straightforward height map, but I found that it is possible to avoid the usual techniques for contouring a 3d scalar field.

This definition outputs both the curves and the meshes. The meshes produced are singly periodic - you can make copies vertically shifted by 2*Pi to get a continuously spiralling surface, and if you also shift them by 1*Pi you get the other half of the helicoids, and it can all be joined into a complete and smooth surface.

So enjoy, I hope you find some interesting and original ways of using and developing this. Please do remember to attribute properly - a lot of effort has gone into this, but I'm freely sharing it in the hope that will be respected.

I've chosen not to compile or obfuscate anything, so you can easily pull it apart and see how it is all working. The original essay linked to at the start contains some suggestions of further reading if you want to learn more about complex numbers and flows.

The file: Rheotomic_Surfaces.gh

Released under the creative commons attribution share alike license 3.0

http://creativecommons.org/licenses/by-sa/3.0/

Comment

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Comment by peyman on April 4, 2014 at 4:14am: Thank you for sharing this awesome definition Daniel ,

the problem for me is , how i can change the frame size for the field as now it is an allocated square

Comment by posfan12 on October 2, 2013 at 2:20pm: Yes, but when you try to save the cluster it does so as a separate file. I.e. the original cluster does not get updated AFAIK.

Comment by Daniel Piker on October 2, 2013 at 6:59am: Hi - I think old clusters don't open just from double clicking, but you can still edit them by right-clicking and choosing "open cluster".

Comment by posfan12 on October 1, 2013 at 11:18pm: One more thing. Why can't I edit the cluster? Is it because it is from an old version of Grasshopper? I am running 0.9.0061.

Comment by posfan12 on October 1, 2013 at 10:45pm: I notice the curve is not centered on the origin. It's off by 1/2 unit in the X and Y directions. Can this be fixed? I know I can move the curve, but then the sources and sinks don't match it.

Comment by Guy Gardner on January 20, 2013 at 5:30pm: Hi Daniel,

The file opens, and the Rheotomic cluster is grey, but the curve and mesh components attached to it are orange. If I open the cluster, the two c# components are red and they both give the error I mentioned.

Comment by Daniel Piker on January 19, 2013 at 6:24am: Hi Guy,

I don't follow when this error occurs - the file opens, and the 'Rheotomic' cluster is there, but orange with no outputs ?

and then you opened the cluster? which C# component throws the error?

It's odd, as it doesn't use any non-standard grasshopper components, and the scripts don't reference any extra libraries.

Comment by Guy Gardner on January 18, 2013 at 9:09pm: Hi, I would like to use this component, I'm not getting errors on opening, but I'm not getting any ouputs and I'm getting the following error from the C# component 1. Solution exception:Could not load file or assembly '0e62753f-604d-4f70-8b8f-6da90af8fe5f, Version=0.0.0.0, Culture=neutral, PublicKeyToken=null' or one of its dependencies. Access is denied.

Any suggestions?

Comment by Anna B on December 19, 2012 at 12:23pm: Tiziano, just to ask the obvious question: maybe you don't have the correct version of grasshopper to open it? Do you get an error message when opening the file? At first I had the same problem so I had to install the last build of grasshopper for Rhino 4. Whichever number that was... Trying to continue opening the file after getting the error will not work.

Comment by Tiziano Derme on December 19, 2012 at 9:49am: hi Guys, I'have downloaded the Rheotomic surface.gh but in the definition the rheotomic component didn't appear.

Could you upload another time, that component please..?

Thanks