Suggestions about Stereographic projection

Question

I have to teach a 1 hr class about the stereographic projection in the complex plane and i am looking for sources or some interesting fact about this. The best I have found is in the Alhfors of Complex Analysis.

It would help me a lot to read your suggestions

Answer 1

A complex number has the classical form a+ib where a,b are real. Non zero complex numbers have the polar form r$e^{i\theta}$ and a third way a complex number can be viewed is, by the stereographic projection as points on the sphere. Each of these representations has a distinct advantage over the other two, depending on the situations. We have to make a judicious choice of the three, to arrive at the desired results.

Answer 2

• Spherical coordinates on $S^2$ correspond to polar or bipolar coordinates on the plane, at the two opposite extremes of where to project from on the sphere.
• There's lots of art and graphics of the Hopf fibration or polychora projected to 3D.
• Stereographic projection is a special case of inversion (circle inversion in 2D or spherical inversion in 3D). Euclidean geometry proofs of conformality and circle-preservingness are possible.
• $\mathrm{SL}_2\mathbb{R}$ acts on the upper half-plane and $\mathrm{SU}(1,1)$ acts on the unit disk. These are two models of hyperbolic geometry (the transformations are the orientation-preserving hyperbolic isometries). The two groups are conjugate within $\mathrm{SL}_2\mathbb{C}$ via the Cayley transform which swaps the half-plane and disk.
• Mobius transformations are sharply 3-transitive, shown with the cross-ratio.
• $\mathrm{SU}(2)$ acts by rotations on the Riemann sphere, exhibiting $\mathrm{SU}(2)/\mathbb{Z}_2\cong\mathrm{SO}(3)$. A higher-dimensional generalization of stereographic projection can turn the Riemann into the projectivized null cone of Minkowski space (the "celestial sphere"), in which Mobius transformations correspond to Lorentz transformations, exhibiting $\mathrm{SL}_2\mathbb{C}\cong\mathrm{SO}(3,1)$.

Stereographic projection of $\Bbb C$, inversion, and Mobius transformations are a package deal.