Circles and lines on the complex plane correspond to circles on Riemann sphere.

Question

I need to prove that circles and lines on the complex plane correspond to circles on Riemann sphere.

This is intuitively easy to understand, as lines on the complex plane can be viewed as a circle going through the point $\infty$. But the thing is, how to describe a circle on Riemann sphere? How to make sure that circles and lines map to circles instead of ellipses or other shapes? So I need some features to determine that the images under this projection are indeed circles.

Any help would be appreciated!

Perhaps start with the intersection of the sphere with a plane being a circle. Three noncolinear points determine a plane, so... — hardmath, Mar 05 '21 at 05:17
@hardmath, Great! It is indeed brilliant to describe circles on Riemann sphere with the intersection of a plane. And I have just found out that this was exactly what Ahlfors did to prove it in his Complex Analysis. Thank you! :-) — atlantic0cean, Mar 05 '21 at 05:21
There is a fascinating connection here with Möbius transformations, but we can leave that topic for another time. — hardmath, Mar 05 '21 at 05:26
Have you heard about stereographic projection (from one of the poles of the sphere to its equatorial plane ? See here or there (fourth figure). — Jean Marie, Mar 05 '21 at 05:45
@JeanMarie When OP said “map,” I presume the map was stereographic projection. Rather vague. — Ted Shifrin, Mar 05 '21 at 05:56
@TedShifrin yeah what I mean here is stereo graphic projection — atlantic0cean, Mar 05 '21 at 06:21

Circles and lines on the complex plane correspond to circles on Riemann sphere.

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