Maps of the sphere!

Question

In the sphere we can introduce two patches that their union covers the whole sphere. Ok, I understand why we need at least two (because of the two poles). The maps in each chart is $\phi_1$ and $\phi_2$ and are defined such that they take the point with polar coordinates $\theta. \phi$ to the points $X,Y$ and $X',Y'$. They go on and say that these maps are the $$\phi_1: X+iY = e^{i\phi}\tan(\theta/2)$$ and $$\phi_2:X'+iY' = e^{i\phi}\cot(\theta/2),$$ and in the overlap we see that $$\phi_2 \circ \phi_1^{-1}(X,Y)=X'+iY'=1/(X+iY).$$ I was wondering how to derive those but I do not see the way. Can you please help me understand why these maps are the way they are?

Answer 1

In your formulas, $\phi$ denotes longitude and $\theta$ denotes angular distance from the south pole, so that $\theta' = \pi - \theta$ is the angular distance from the north pole.

Write complex numbers in polar form $z = \rho e^{i\phi}$, and consider the longitudinal cross section below. Your formulas come from trigonometry, noting that $$ \rho = \tan(\theta/2) = \cot(\theta'/2). $$

In a bit more detail, one coordinate chart is induced by sterographic projection from the north pole. The other chart is the complex conjugate of stereographic projection from the south pole.

The conjugate is omitted in your formulas, but is needed because $$ \frac{1}{z} = \frac{1}{\rho} e^{-i\phi}. $$ (This oversight is not uncommon even in the mathematics literature.)