How to describe the relative positions?

Question

There is a problem in my complex variable textbook as follows:

Discribe the relative positions of the images of $z$, $-z$ and $\bar z$ on the Riemann sphere.

But I don't understand what does this mean. My thought is as for the point $z=x+iy$ on the complex plane the related points on the Riemann sphere is

$$x_1=\frac{2(z+\bar z)}{|z|^2+4},x_2=...,x_3=...$$

we just need to put $-z$, $\bar z$ instead of $z$ in these three formulas. But I'm not sure if this is what the question has required. So could someone clarify this question for me? Thanks!

Answer 1

Assuming that, for you, the connection between $\mathbb C$ and the Riemann sphere$$S^2=\{(a,b,c)\in\mathbb{R}^3\,|\,a^2+b^2+c^2=1\}$$is given by the stereographic projection$$\begin{array}{ccc}S^2\setminus\{(0,0,1)\}&\longrightarrow&\mathbb C\\(a,b,c)&\mapsto&\dfrac{a+bi}{1-c},\end{array}$$then the answer is:

• $z$ and $\overline z$ are the reflection of each other with respect to the plane $b=0$;
• $z$ and $-z$ are the result of applying a half-turn around the $z$-axis to each other.