I'm trying to find an explicit bijection from $\{(x,y):(x^2+y^2=1)\}$ and the real numbers $\mathbb R$.
I've found a couple bijection to go from $\mathbb R$ to the unit circle, but none that go the other direction. Any help would be greatly appreciated.
Best Answer
As commenters said, there is no nice bijection. Conceptually this is the same story as with bijections between closed and open intervals.
There is a natural bijection from the unit circle minus the point $(-1,0)$ to real line: $(x,y)\mapsto y/(x+1)$. Geometrically this is radial projection:
Then you have to create room for $(-1,0)$. Use Hilbert's hotel: on the real line, send $n$ to $n+1$ for $n=0,1,2,3,\dots$. Then $(-1,0)$ can be mapped to $0$.