complex anaylsis : stereo-projection

Alright. The complex plane is essentially your old, run-of-the-mill xy-coordinate system. Now, imagine that a sphere of radius ½ is resting above the complex plane (meaning we how have a z-axis which passes through (0,0), is perpendicular to the x- and y- axis, and follows the right-hand rule) with its center point at (0,0,½).

Now, pick an aribtrary point P(x,y) on the complex plane. Connect the point Q which lies at (0,0,1) (=top of the sphere) with P(x,y) with a straight line. There will be exactly one point on the sphere (other than Q) which will lie on this connecting line. Let's call it S(x,y,z).

For another choice of P(x,y), say P'(x,y), there will be another S, S'(x,y,z). The mapping is 1-1. The entire complex plane (or xy-plane if you will) is now contained on the surface of this sphere.

Furthermore, when a line is made from Q(0,0,1) parallel to the plane it does not intersect the sphere. This line can be imagined to be going to infinity, in which case Q(0,0,1) becomes the map for infinity. This sphere then contains a single point which is infinity.

Stereo-projection is interesting for several reasons. One, it shows why in complex analyis, lines are circles (there is no differentiation between the two.) (Hold on, imagining this may get a little difficult) Take a circle on the sphere which does not pass through Q. It will map onto a circle on the complex plane. This is fairly straightforward. However, take a circle on the sphere which passes through Q (remember Q is infinity.) In order for the map of the circle on the sphere to go to infinity, it can only be a straight line. In the complex plane, circles and lines are really just representations of one and the same object on this sphere.

And probably the other most interesting result is that having a single infinity rather than an infinite number of infinities (i.e. uncountably infinite) allows for many calculations, which I don't think we need to go into right now.

Yeah, so there's stereo-projection. Let me know if it's not clear. -fb

for background information see Mathematics Metanode