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Stereographic Projection, Riemann Sphere and Mapping

A plane is inserted through the equator of a unit sphere.
A point on the sphere is mapped onto the plane by creating a line from the point on the sphere, through the north pole, where this line hits the plane is the projection of the point.
Show that circles on the sphere map to circles on the plane except when the circles runs through the north pole, in this case it maps onto a line.


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A Stereographic Projection

Consider a sphere of radius 1, with a plane inserted through the equator

We map the sphere onto the plane by mapping a given point P onto Q, such that P, Q and the North Pole N all lie on the same line. Given this mapping the southern hemisphere maps inside the circle of radius 1, the equator stays where it is and the north hemisphere maps onto the region outside the unit circle. Note that the North Pole maps onto "infinity".

Let's say the point P has coordinates (x,y,z), Q lies at (a,b,0) and N at ...

Solution Summary

Stereographic Projection, Riemann Sphere and Mapping are investigated. The solution is detailed and well presented. Diagrams are included.