I'm finally working my way through Sarason's Complex Function Theory.
The mapping from the complex plane into the sphere has the rule: $$ z \mapsto \frac{(2 \operatorname{Re} z,2\operatorname{Im}z, |z|^2-1)}{|z|^2+1}. $$ Let $\rho(z_1,z_2)$ denote the Euclidean distance between two points on the sphere. Show that the corresponding metric on the complex plane $\mathbb{C}$ is: $$ \mu(z_1,z_2)=\frac{2|z_1-z_2|}{\sqrt{|z_1|^2 +1}\sqrt{|z_2|^2 +1}} $$ I think I could establish this equivalence by using: $$ z_1=x_1+iy_1\\ z_2=x_2+iy_2 \\ |z_1-z_2|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2} $$ and lots of algebra. Is there a simpler way to establish this metric?
It seems like it's easy enough to show $$ \mu(z_1,z_2)\ge 0, \text{with equality if }z_1=z_2 \\ \mu(z_1,z_2)=\mu(z_2,z_1)\\ \mu(z_1,z_2) + \mu(z_2,z_3) \ge \mu(z_1,z_3) $$ Still, I'd like to understand more deeply what it means for the two metrics to be equivalent.