If you operate the stereographic from spherical coordinate to polar coordinate in the complex plane you obtain the well known relation $$z=\cot\frac{\theta}{2}e^{i\phi}.$$ I'd like to know if there's a pictorial and simple way to visualize this. Clearly $\phi$ is the same in spherical coordinate and polar coordinate so the picture is clear. I thought that also the cotangent would be clear but what bothers me it's the fact that the modulos of the complex number is given by $\frac{\theta}{2}$ instead of $\theta$.
I know algebrically when it comes from, I'd like to know if there's a pictorial way to see where this factor $1/2$ comes from.
Such a pictorial explanation for the transformation
$$\theta \to \frac12 \theta \tag{1}$$
is possible and even easy.
Consider Fig. 1, which represents the vertical section of the unit sphere passing through its vertical axis and current points $P$ and its stereographic projection $P'$.
Fig. 1. $\theta$ is the colatitude of $P$. $P'$ is the image of $P$ by the South pole stereological projection.
Transformation (1) is due to the inscribed angle theorem (https://en.wikipedia.org/wiki/Inscribed_angle).
More precisely what we obtain is
$$|z|=OP'=\tan\left(\frac\theta2\right)\tag{2}$$
(I beg your pardon : it was simpler to explain it with colatitude instead of latitude : this is why we obtain $\tan$ in (2) instead of cotan).