The geometry of complex projective coordinates

After my recent discussion of Möbius transformations as the "other" bilinear mappings, I was asked if there's a geometric meaning for projective representations of complex numbers. Here's the best answer that I could come up with. It concerns relating the projective coordinates to points on the Riemann sphere

So suppose that we have two complex numbers z and w that have projective representations

z = z1 / z2

and

w = w1 / w2

that we write as

z =
 
z1
z2
 

and

w =
 
w1
w2
 

Now suppose that z and w are orthogonal. This means that their inner product is zero, or that 

(z,w) = z*w = z1*w1 + z2*w2 = 0

This means that

z1*w1 =  -z2*w2

or

w1 / w =  -z2* / z1*

or

w = -1 / z*

But there's a nice geometric interpretation of that. In particular, the points z and -1/z* are antipodal (diametrically opposed) on the Riemann sphere. So if the projective representations of two complex numbers are orthogonal, then the two complex numbers correspond to antipodal points on the Riemann sphere.

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