Exercise [08.06]

variounes · 09 Feb 2009, 20:37

Prove that the two stereographic projections in Figs 8.7a,b are related by $w = z^{-1}$ .
$z = r e^{i\theta}$
$\frac{1}{z} = \frac{1}{r}e^{i(-\theta)}$
$w = d e^{i\theta'}$
Two things to be proved: $\theta' = -\theta$ and $d = \frac{1}{r}$
The former is obvious from the directions of the imaginary axis in the two figures. The latter is a fascinating exercise, involving elementary-school geometry stuff, i.e. circles, angles and triangles. An elegance solution is to realise that in Fig 8.7b the two triangles {O,N,r} and {O,d,N} are similar, with N and O being the north pole and the sphere's origin. (Note that both triangles are right-angled). Therefore, the distance ratios:
$\frac{ON}{r} = \frac{d}{ON}$
On the other hand, ON = 1

(unit sphere's radius).
$d = \frac{1}{r}$

Exercise [08.06]

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