Find the sets of points on the sphere xi^2+eta^2+zeta^2=1…
Find the sets of points on the sphere $\xi^2+\eta^2+\zeta^2=1$ whose stereographic projections satisfy $\ 1^\circ\ \arg z=\alpha;\quad2^\circ\ |z|=r$.
Explanation of Method
The ray $\arg z=\alpha$ is exactly the image of a single meridian (Problem 105), so its preimage is that semi-meridian. The circle $|z|=r$ has constant $|z|^2=r^2$, and the inverse formula for $\zeta$ (Problem 104) depends only on $|z|^2$ — so the preimage is a full parallel (horizontal circle) at the corresponding constant latitude.
Case $1^\circ$: $\arg z=\alpha$
Step 1 · Recall the meridian result
By Problem 105, the meridian at longitude $\alpha$ — parametrized by $(\xi,\eta,\zeta)=(\cos\theta\cos\alpha,\cos\theta\sin\alpha,\sin\theta)$ for $\theta\in\left[-\frac\pi2,\frac\pi2\right)$ — projects exactly onto the ray $\{re^{i\alpha}:r\ge0\}$.
Step 2 · Match to the ray $\arg z=\alpha$
The condition $\arg z=\alpha$ (for $z\neq0$) describes precisely this ray minus the origin; including $z=0$ (which has $\arg z$ conventionally undefined, but is the natural endpoint), the preimage is:
$$ \boxed{(\xi,\eta,\zeta)=(\cos\theta\cos\alpha,\ \cos\theta\sin\alpha,\ \sin\theta), \qquad \theta\in\left[-\tfrac\pi2,\tfrac\pi2\right)} $$
the semi-meridian from the south pole (inclusive, $\theta=-\frac\pi2$) up to (but not including) the north pole.
Case $2^\circ$: $|z|=r$
Step 3 · Use the inverse formula for $\zeta$
From Problem 104: $\ \zeta=\dfrac{|z|^2-4}{|z|^2+4}$. Since this depends only on $|z|^2$, every point with $|z|=r$ shares the same:
$$ \zeta=\frac{r^2-4}{r^2+4} \quad(\text{a constant}) $$
Step 4 · The corresponding latitude circle's radius
Points at fixed $\zeta$ on the unit sphere form a horizontal circle of radius $\rho=\sqrt{1-\zeta^2}$. Compute:
$$ 1-\zeta^2=1-\left(\frac{r^2-4}{r^2+4}\right)^2=\frac{(r^2+4)^2-(r^2-4)^2}{(r^2+4)^2}=\frac{16r^2}{(r^2+4)^2} $$
$$ \rho=\frac{4r}{r^2+4} $$
Step 5 · Conclusion
$$ \boxed{\xi^2+\eta^2=\left(\frac{4r}{r^2+4}\right)^2, \qquad \zeta=\frac{r^2-4}{r^2+4}} $$
a full circle (parallel) at constant latitude, since as $\arg z$ ranges over all of $[0,2\pi)$ with $|z|=r$ fixed, every longitude is covered.
Special cases
| $r$ | $\zeta$ | Description |
|---|---|---|
| $0$ | $-1$ | South pole (degenerate circle) |
| $2$ | $0$ | Equator |
| $\to\infty$ | $\to1$ | Approaches north pole |
Result
Polar coordinates in the plane correspond exactly to spherical coordinates on the sphere: constant argument (rays) map to meridians, constant modulus (circles centered at $0$) map to parallels.