The latest Kangaroo (for Rhino 6 only) includes a new component for Möbius transformations.
There are some notes to explain bit about how this works and what it can be used for here:
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Here’s an example definition showing how to generate the Möbius band with circular boundary:
mobiuscircular.gh (15.5 KB)
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Hi Daniel,
I’m familiar with Mobius transformations in the complex plane but have never seen a 3 dimensional application (except for stereographic projection) as you are performing here. Is there any literature you’d recommend to understand what’s going on in your component?
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Did you check the Google doc he attached?
With all due respect, that really doesn’t open up the subject for me. Maybe I’ll regret it, but I could really use something with more meat (math) on it.
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No worries, was only asking because I did actually miss it at first (was too mesmerized by the gif). 
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Absolutely, there’s a lot of great material on this topic, and my doc is only the briefest of introductions. I’ll add a bigger links section later, but some that come to mind are:
this 9 part video series (available in many languages):
http://www.dimensions-math.org/
Here’s another shorter but nice one I saw just the other day, exploring this through VR:
The page that did most to first get me interested in the topic was this one by Thomas Banchoff:
For something drier maybe this book(pdf)
And of course the beautiful book Visual Complex Analysis by Needham, which covers many other topics, but includes some material on connections between Möbius transformations and rotations
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Thanks Daniel, I’ll look into them.
Another example showing transformations of a relaxed compact circle packing and use of the FixSphere option:
Circlepack_transformation.gh (12.4 KB)
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Steiner Circles using this component.
Steiner Circles.gh (19.6 KB)
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Nice!
by the way - you can input circles to the geometry component directly (you don’t have to pass it sets of 3 points and reconstruct the circle after)
Good to know; I assumed something would get lost in translation. I figured I’d give it a try with spheres then and that also works. I guess you’re looking to see whether its a circle or a sphere and just extracting the center and radius values.
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An old video about this topic
My first attempt to apply Möbius transformations on a 2D geometry using grasshopper:
Möbius transformations.gh (472.8 KB)
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I don’t know what the purpose of using Möbius transformations ;but i try this based on the video posted by @Mahdiyar
Mobius transf.gh (16.3 KB)
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Interesting, but those aren’t Möbius transformations - notice how the angles are distorting - this also means it wouldn’t preserve circles.
Stereographic projection of points in 3d from an origin centred sphere to the plane can actually be done very simply:
The component described in this post uses the equivalent projection in 4d - first inverse stereographic projection to take everything to the curved 3d space of the 3-sphere, then a rotation in 4d space, then stereographic projection back to our flat 3d space.
It also works not just on points, but all types of geometry, including NURBS surfaces, which generally requires not just a simple transformation of the control points, but a complete rebuilding including changing the point count, degree and trim curves to give an accurate transformation of the surface itself:
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just watched this video talks about transformation at about 10 min in.
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https://www.grasshopper3d.com/photo/inscribed-rectangle-proof?context=latest
https://www.grasshopper3d.com/forum/topics/inscribed-rectangle-proof-samples-examples
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The same video maker has a good one about quaternions and stereographic projection too:
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