New version of an interactive program f

New version of an interactive program for exploring Möbius transformations. N...
Tim HuttonTim Hutton - 2016-01-15 21:45:04+0000 - Updated: 2016-01-15 22:05:04+0000
New version of an interactive program for exploring Möbius transformations. Now with properly curved grid lines and arrows.

Question: is the resemblance of the image below to vortices you might see in your cup of coffee just a pleasing coincidence, or is there something about smooth transforms of sufficient complexity that leads to what looks like turbulence?


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Roice Nelson - 2016-01-16 22:08:44+0000 - Updated: 2016-01-16 22:14:30+0000
The arrows are really effective!

I bet the latter chapters of Visual Complex Analysis would help answer your question, because they tie Möbius transformations to physical systems.

I do know the general case has particle paths that are loxodromic. The following video is a cool example that I think exhibits the behavior of elliptic Möbius transformations with two fixed points, perhaps similar to patterns you've seen in coffee swirls.
Gerard Westendorp - 2016-01-17 10:06:45+0000
The real and imaginary part of a complex function are solutions to the Laplace equation. If you solve a potential flow from the poles to the zero’s, then lines orthogonal to the potential flow are real and imaginary parts oa a complex function.
These will look as vortices, because these are also solutions to the Laplace equation. I guess your arrows would be a complex function, ie the arrow (dx +idy) are f(x+iy).
Just thinking:
Maybe instead of lines orthogonal to a potential flow, you could construct a viscous flow directly that emulates the arrows. Instead of pumping fluid in and out at the poles and zero’s, you put shafts (axis orthogonal to the complex plane) rotating counterclockwise at the poles, and clockwise at the zero’s, and solve the viscous flow.
That might be fun..

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