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Simple question: Are there stereographic projections from the circle on the to x-y plane? If yes, how are they formulated? I've read the nice article on wikipedia and it mentions stereographic projections from the sphere onto the plane. I was wondering if the projection could be restricted to the X-Y plane.


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up vote 2 down vote accepted

Yes, usually a stereographic projection takes a sphere (minus a point) to the plane, but the principle generalizes straightforwardly to other dimensions. It then projects the $n$-sphere minus a point to $\mathbb R^n$.

For example you can restrict the usual stereographic projection to a (great or small) circle containing the missing point, which will give you a projection from a circle missing one point to an infinite line. It will be something like $$p: (-\pi, \pi)\to\mathbb R, \qquad p(\theta)=\tan(\theta/2)$$ where the argument to $p$ is a point on the circle expressed in radians.

If you restrict the 2-dimensional projection to a circle on the sphere that doesn't contain the antipodal point, it projects into a circle or ellipse in the plane, which isn't much fun. (If the circle is a latitude line, you gain nothing more than a linear scaling).

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Thanks Henning... – titi Oct 4 '11 at 23:22

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