The Riemann Sphere

The Riemann sphere is the geometric representation of the extended complex planeobtained by placing a unit spehre at the origin. For each point in the plane, a line is drawn from that point to the North Pole. The line will intersect the sphere at a single, unique point. We can represent the pointin the natural way as the North Pole.

The projection is one to one and onto, so functions in the complex plane can be transformed into functions on the Riemann sphere. For example, any rational function on the complex plane can be extended to a continuous function on the Riemann sphere, with the poles of the rational function mapping to infinity. 

Ifis a point in the plane, it projects to the point

Conversely a pount (u,v,w) on the sphere satisfies

The inverse transformation can be easily derived.

Consider the component of the inverse transformation along the real axis.

Consideration of similar triangles gives similarly for

and

To find P(x,y), we write

Now

Also

Hence