The following two distance functions can be used to compute the distance between points on the stereographic projection of the sphere:
$$ d(x,y)=\frac{1}{\sqrt{K}}\arccos\left(1-\frac{2K||x-y||_2^2}{(1+K||x||_2^2)(1+K||y||_2^2)}\right) $$
and
$$ d(x,y)=\frac{2}{\sqrt{K}}\arctan\left(\sqrt{K}||(-x)\oplus_K y||_2\right) $$ where $\oplus_K$ represents the Möbius addition:
$$ x\oplus_Ky = \frac{(1-2K\langle x,y\rangle -K||y||_2^2)x + (1+K||x||_2^2)y}{1-2K\langle x,y\rangle + K^2||x||_2^2||y||_2^2} $$
I've tested the formulas and numerically they yield the same results.
Can you show that they are equivalent?
