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

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