Stereographic Projection

A fairly common alternative for parameterizing 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.

On the three-dimensional sphere $S^2$, it is possible to define a function $\varphi_{+3}$ as a stereographic projection from $U_{+3} := S^2 /\ \left[0,0,1\right]^{\top}$ to $\mathbb{R}^{2}$:

$$\begin{array}{c} \varphi_{+3} : U_{+3} \mapsto \mathbb{R}^... ...ac{1}{1-z} \begin{bmatrix} x \\ y \end{bmatrix} \end{array}$$ (1.56)

together with its inverse

$$\varphi_{+3}^{-1} \left( \left[u,v \right]^{\top} \right) =... ... \begin{bmatrix} 2u \\ 2v \\ -1 + u^2 + v^2 \end{bmatrix}$$ (1.57)

where $\left[0,0,1\right]^{\top}$ denotes the “north pole” of the sphere. $\varphi_{+3}$ and $\varphi^{-1}_{+3}$ are continuous on $U_{+3}$ and therefore the stereographic projection is a homeomorphism. This projection establishes a relation between the point $(u,v,0)$ on the plane $z=0$ and the point on the sphere $(x,y,z)$, given by the intersection between the projection ray joining the origin $(0,0,1)^{\top}$ (the only singular point) to the point on the plane and the unit-radius sphere, as shown in Figure 1.3.

Similarly, spaces $U_{\pm i} := S^2 /\ \pm e_i$ can be defined in which the $e_i$ are unit vectors, within which six similar parameterizations can be defined (each with its own singularity). This makes it possible to choose the most appropriate parameterization so as to operate at the point farthest from the singularity of that particular formula.