On the unit sphere xi^2+eta^2+zeta^2=1 in Oxietazeta -space…
On the unit sphere $\xi^2+\eta^2+\zeta^2=1$ in $O\xi\eta\zeta$-space, project each point $P=(\xi,\eta,\zeta)$ (other than the north pole $N=(0,0,1)$) from $N$ onto the plane $\zeta=-1$, identified with $\mathbb C$. Find the resulting map $z=f(\xi,\eta,\zeta)$.
Explanation of Method
Parametrize the line through $N$ and $P$, find the parameter value at which it hits the plane $\zeta=-1$, and read off the $x,y$ coordinates of the intersection point there.
Step 1 · Parametrize the projecting line
The line through $N=(0,0,1)$ and $P=(\xi,\eta,\zeta)$ is:
$$ \mathbf r(t)=(0,0,1)+t(\xi,\eta,\zeta-1)=\big(t\xi,\ t\eta,\ 1+t(\zeta-1)\big) $$
Step 2 · Find where the line meets $\zeta=-1$
$$ 1+t(\zeta-1)=-1 \quad\Longrightarrow\quad t=\frac{-2}{\zeta-1}=\frac2{1-\zeta} $$
Step 3 · Read off the intersection coordinates
$$ x=t\xi=\frac{2\xi}{1-\zeta}, \qquad y=t\eta=\frac{2\eta}{1-\zeta} $$
Step 4 · Express as a complex number
$$ \boxed{z=x+iy=\frac{2(\xi+i\eta)}{1-\zeta}} $$
Step 5 · Special points
| Point on sphere | $z$ |
|---|---|
| North pole $(0,0,1)$ | $\infty$ (by convention) |
| South pole $(0,0,-1)$ | $\dfrac{2(0)}{2}=0$ |
| $(1,0,0)$ | $\dfrac{2(1)}{1}=2$ |
| $(0,1,0)$ | $\dfrac{2i}{1}=2i$ |
| $(-1,0,0)$ | $\dfrac{-2}{1}=-2$ |
Verification
Every point on the sphere with $\xi^2+\eta^2+\zeta^2=1$ satisfies $\xi^2+\eta^2=1-\zeta^2=(1-\zeta)(1+\zeta)$. Computing $|z|^2$ from Step 4: $\ |z|^2=\dfrac{4(\xi^2+\eta^2)}{(1-\zeta)^2}=\dfrac{4(1-\zeta)(1+\zeta)}{(1-\zeta)^2}=\dfrac{4(1+\zeta)}{1-\zeta}$, which is finite and well-defined for every $\zeta\neq1$, confirming the map is defined on the whole sphere minus the north pole.
Result
This is the stereographic projection formula for a sphere of radius $1$ projected onto the plane $\zeta=-1$ (rather than the more common convention of projecting onto the equatorial plane $\zeta=0$, which would give $z=\dfrac{\xi+i\eta}{1-\zeta}$ — exactly half of this map). It is conformal, maps circles on the sphere to circles or lines in the plane, and sends the north pole to the point at infinity.