This is question 5 on page 20 of the book Complex Analysis by Lars Ahlfors. I have no idea how to answer that problem:
Find the radius of the spherical image of the circle in the plane whose center is $a$ and radius is $R$.
Here spherical image means: the image of a subset of complex numbers under the identification of the complex plane with the sphere $\Bbb S^2$ (the Riemann sphere) by stereographic projection:
Thanks.
Best Answer
As you can see from the picture, the geometry is symmetric about the vertical axis, in other words the answer depends only on $R$ and $A=|a|$. So if we take our circle to be centered on the $x$-axis, as @Hagen has suggested, we only need to look at the intersection of the picture with the $(x,z)$-plane. The map from the $x$-axis to the circle $x^2+z^2=1$ is: $$ \xi\mapsto \left(\frac{2\xi}{\xi^2+1},\frac{\xi^2-1}{\xi^2+1}\right)\,. $$ Now you have to plug in $A+R$ and $A-R$ for $\xi$, find the two points the formula gives you, and the distance between them is the diameter of the circle you want, again making use of @Hagen’s suggestion. Looks like very messy algebra, and I do wonder whether there’s a slicker way to do it.
I suppose I should boast that I learned this stuff out of Ahlfors’s book, with Ahlfors himself standing in front of the class. He was a superb teacher.