Question

3. If Z and W are two distínct points on the Riemann sphere, then the plane through these points and the origin cuts the sphere in a "great circle," that is, a circle with maximum diameter (2, for a unit sphere). Show that this great circle corresponds to the unique circle (or line) in the plane that passes through the points z, w, and 1/z, where Z, W are the projections of z, w respectively. [HINT: See Prob. 2.]

Answer 1

Let be the point on the Riemann sphere.

Z =(21, 22.23

The projection of Z is
21iz z = 1 z3
. And
1/-1iz2) 8
where
a23-1,8- ,A 8
.

After $\frac{1}{\overline{z}}$ back to the Riemann sphere we have to show that it lies on the great circle. In other words it is enough to show the it lies on the plane spanned by the origin,Z and W.

If we show that it is a linear multiple of Z we are done.

The projection of $\frac{1}{\overline{z}}$ back to the Riemann sphere is given by . (Steriographic projection)

-1 + Η 2λ21 2λ2 _ 1+4+ Σ 2 Ν

We just have to show that $\widetilde{Z}$ is a linear multiple of Z, so we can ignore any scalar we multiply to $\widetilde{Z}$

Which reduces $\widetilde{Z}$ to

) 2A1, 2A22,-1 2

2.21 (2aB21, 2aB22,-Ba az

(2aB21, 2a2, (25)((2-1)-2))

$\equiv (2 \alpha z_1, 2 \alpha z_2,z_3^2 -2 z_3 +1 -z_1^2-z_2^2)$

But (As Z lies on a unit sphere)

$\equiv (2 \alpha z_1, 2 \alpha z_2,2z_3^2 -2 z_3) \equiv (2 \alpha z_1, 2 \alpha z_2,2\alpha z_3) \equiv (z_1,z_2,z_3)$.

Hence $\frac{1}{\overline{z}}$ lies on the circle.