What is the geometric way of relating zero to infinity?

Question

I once saw (what I think was) a geometric way a relating zero to infinity. Something about a circle with radius 1 around the origin. Can you tell me where to find that?

Thanks

Answer 1

In the Riemann sphere, the complex plane augmented with a point at infinity, you can make the substitution $w=\frac 1z$. Each point is "reflected across the unit circle" and $\infty \leftrightarrow 0$