Consider Riemann sphere
And consider the following projection :
The plane $\zeta =0 $ here is the complex plane, and consider the following map:
Each point (except the north pole) of the sphere, being mapped to $ \left(\xi,\eta,\zeta\right)\to\left(\frac{\xi}{1-\zeta},\frac{\eta}{1-\zeta},0\right) $
And conversly, the point on the sphere that is being mapped to $(x,y,0) $ given by
$$ \left(\frac{x}{x^{2}+y^{2}+1},\frac{y}{x^{2}+y^{2}+1},\frac{x^{2}+y^{2}}{x^{2}+y^{2}+1}\right)\to\left(x,y,0\right) $$
Now, I want to show that the function $ \frac{1}{z},z\in\mathbb{C} $ is represented on the sphere by a $ 180^{\circ} $ rotation about the diameter with endpoints $ \left(-\frac{1}{2},0,\frac{1}{2}\right),\left(\frac{1}{2},0,\frac{1}{2}\right) $
What I have done :
I proved that given a point $ z $ which is the image of a point $ \left(\xi,\eta,\zeta\right) $ on the sphere, the point which is mapped to $\frac{1}{z} $ on the sphere, is the point $ \left(\xi,-\eta,1-\zeta\right) $.
So all thre's left to do is to prove that given a point on the sphere $ \left(\xi,\eta,\zeta\right) $, its rotation about the diameter that I mentioned by $180^{\circ} $ is indeed $ \left(\xi,-\eta,1-\zeta\right) $.
I dont know how to express the rotation since its in 3 dimensions and Im not familier with this. So I'd really apreciate any help.
Thanks in advance.
With the transformation of the coordinate system $$ \begin{align} \xi &= u \\ \eta &= v \\ \zeta &= w + 1/2 \end{align} $$ the mapping $$ \begin{pmatrix} \xi \\ \eta \\ \zeta \end{pmatrix} \mapsto \begin{pmatrix} \xi \\ -\eta \\ 1-\zeta \end{pmatrix} $$ becomes $$ \begin{pmatrix} u \\ v \\ w \end{pmatrix} \mapsto \begin{pmatrix} u \\ -v \\ -w \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0& -1 \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix} $$ and that is a rotation about the $u$-axis by $180$ degrees, compare basic 3d rotations.
So the original mapping is a rotation by $180$ degrees, and the rotation axis is the $\xi$-axis translated by $1/2$ in the direction of the positive $\zeta$-axis (which is what you wanted to show).