Stereographic Projection

A fairly common alternative to parameterize the sphere $S^n$ is to use stereographic projection to transform coordinates from the manifold space $\mathbb{R}^{n}$ (parameter space) to $\mathbb{R}^{n+1}$ (Cartesian space) and vice versa.

Figure 1.3: Stereographic projection.

In the three-dimensional sphere $S^2$, a function $\varphi_{+3}$ can be defined as the stereographic projection from space $U_{+3} := S^2 /\ \left[0,0,1\right]^{\top}$ to $\mathbb{R}^{2}$:

$$\begin{array}{c} \varphi_{+3} : U_{+3} \mapsto \mathbb{R}^... ...ac{1}{1-z} \begin{bmatrix} x \\ y \end{bmatrix} \end{array}$$ (1.20)

along with its inverse

$$\varphi_{+3}^{-1} \left( \left[u,v \right]^{\top} \right) =... ... \begin{bmatrix} 2u \\ 2v \\ -1 + u^2 + v^2 \end{bmatrix}$$ (1.21)

where $\left[0,0,1\right]^{\top}$ denotes the "north pole" of the sphere. $\varphi_{+3}$ and $\varphi^{-1}_{+3}$ are continuous in $U_{+3}$, thus the stereographic projection is a homeomorphism. In this projection, a relationship is established between the point $(u,v,0)$ on the plane $z=0$ and the point on the sphere $(x,y,z)$, the intersection between the projective line connecting the origin $(0,0,1)^{\top}$ (the only point of singularity) with the point on the plane and the unit sphere, as shown in figure 1.3.

Similarly, spaces $U_{\pm i} := S^2 /\ \pm e_i$ can be defined where the $e_i$ are the unit vectors within which to define 6 similar parameterizations (each with its own distinct singularity), allowing the selection of the most appropriate parameterization to operate at the point furthest from the singularity of that specific formula.