The Riemann Sphere

When explaining the Riemann sphere, think of the complex plane Z as an endless 2D plane. The non-existent “edges” of the plane can be considered as infinity (mf344, 2013). To help visualize this, I imagine walking from the center of a never-ending piece of paper in any direction towards a non-existent edge representing infinity like in figure 2. Now imagine that the paper is then folded so that it somewhat resembles a circle, where all of its edges (infinities) meet at one point. In S, this one point is (0,0,1) and is called the north pole, N, also called infinity, ∞.

Since the circle is originally formed from the complex 2D plane, each point on the sphere represents a point on the 2D plane; therefore it’s a one to one mapping (mf344, 2013). This is true for every point except for point ∞(explained further on). So if we were to combine the sphere and a complex plane together so that the plane intersects the equator of the sphere at c=0, we would end up with the Riemann Sphere, where every point on the sphere

This equation gives us the point P(A) on sphere S corresponding to point A on the complex plane Z. The general equation for finding point (a,b,c) on S corresponding to point (x,y) on Z which is given being

(Proof 2)
Another way of looking at line L would be by connecting point ∞ to point P(A) in figure 3. Line L can be defined as:

(University of Oxford, 2012)

This is the same line L as in the first proof but in a different perspective that will help in explaining the next proof.

The vector v, (a¦█(b@c-1)) is the vector used to move from point ∞ to point P(A). This can be found by subtracting the coordinates of point P(A) by the coordinates of point ∞.

When line L is extended and intersects the complex plane Z, this point will be point A.

Since Z is a xy plane, the z value of point A must be z=0, since point A is a point on the xy plane.