Let $S^2\subset\mathbb{E}^3$ be a sphere with radius 1 and center $(0,0,1)$ in cartesian coordinates. The northpole is point $(0,0,2)$ on the sphere and $Oxy$ is the xy-plane. Let $\pi:S^2\setminus N \to Oxy $ be a stereographic projection defined by $\pi(x,y,z)=(u,v)$ where $(u,v)$ is the intersection of the xy-plane and of a straight line through the the north pole and point $(x,y,z)$ on the sphere. I know $\pi$ is given by \begin{equation} \pi(x,y,z)=(u,v)=(\frac{2x}{2-z},\frac{2y}{2-z}), \end{equation} and its inverse $\pi^{-1}:\mathbb{R}^2\to S^2\setminus N$ by \begin{equation} \pi^{-1}(u,v)=(\frac{4u}{u^2+v^2+4},\frac{4v}{u^2+v^2+4}, \frac{2(u^2+v^2)}{u^2+v^2+4}) \end{equation} The sphere can be parametrized by \begin{equation} r(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta). \end{equation} Now I am trying to understand what exactly the metric induced on the $Oxy$ plane under the stereographic projection means. Is it a way to measure distance from between points on the sphere through their projections on $\mathbb{R}^2$? How would you go about such a computation? Thank you.
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