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@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the sphere. When do we use which? In particular when we are using the projections to investigate corresponding Möbius maps and cross ratios etc, since clearly they have different effects. For the second definition where is the "source of projection"? Thank you.

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Up to some constant multiplicative factors floating around, the various definitions of stereographic projections are all equivalent. In particular they have the same "set of Mobius maps" since Mobius transformations are closed under function composition, and the difference between the two stereographic projections is a scaling by a factor of 2, which is a Mobius transformation. – Willie Wong Feb 9 '12 at 12:05
One is merely a rescaling of the other. You can put the plane anywhere below the north pole, and it's just a rescaling of what you'd get if you use any other plane parallel to it below the north pole. And you can put it above the north pole if you don't mind the scale factor being negative (so you'd have an orientation-reversing mapping instead of an orientation-preserving mapping. – Michael Hardy Feb 9 '12 at 16:14

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