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I have been battling with this problem for some time now but I am still stuck. Please would some kind soul help me out?

So I have a Riemann sphere with 2 point $X,Y$ on it. (The Riemann sphere is just a unit sphere right?) The spherical distance between $X,Y$ is $l$. The projection point is equidistant from $X,Y$. I have to find the stereographic projection of $X,Y$, where we are told that the projections lie on the real axis.

My argument: the angle $X$ makes with the axis normal to the plane $\mathbb C$ would be $l\over 2$. So the "projection line" radiating from the projection point, passing through $X$ and intersecting $\mathbb R$ makes an angle $l\over 4$ with the vertical. So the distance of the projection from the origin $0\in\mathbb C$ would be $2\tan({l\over 4})$.

The given answer: The distance of the projection from the origin $0\in\mathbb C$ is $\tan({l\over 4})$

I don't understand why there isn't a factor of $2$ since if the radius of the sphere is $1$, surely the diameter has length $2$??

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    @WillieWong: Thank you very much!2012-02-09

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