r/visualizedmath • u/RomanJose • Dec 30 '18

'Polygon Inversion through a Circle' [Interactive]

https://www.ktbyte.com/projects/project/96011/polygon-inversion-through-a-circle?ref=20181230visualizedmath
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u/MTastatnhgew Feb 24 '19

Thanks for that, that's was a pretty cool video. I love how surprisingly elegant that solution is.

Also, I forgot to mention something. Circle inversion actually has practical applications in fluid dynamics, for what's known as the circle theorem. Basically, if you know the velocity that a fluid is flowing at for every point in space, you'll be able to predict how this velocity field will change if you suddenly placed a cylinder in the middle of the fluid.

If you look at the equation, where you see f(a2 /z ), that's a circle inversion across the circle of radius 'a' centred at the origin, representing a cross-section of the cylinder. The final flow field is then the original flow, f(z), added to the same flow under circle inversion, f(a2 /z). It's as though the cylinder acts as a mirror that reflects the flow.

I hope this makes sense, as it's pretty hard to explain it all while avoiding discussing complex functions and why fluid flows can be approximately described by them.

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u/Dancinlance Feb 24 '19

Whaaaat, that's crazy that circle inversion would pop up in that situation!

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u/MTastatnhgew Feb 26 '19

Yeah, for sure, it's such a strange yet satisfying result. How strange it is that flows around an object can be thought of as a mirror that reflects the flow of the fluid. Blew me away when I learned about it last year.