The following question came to me while programming some visualizations for Mobius Transformations as a pet project to learn Mathematica.
Generally, the inverse of a sphere under sphere-inversion is also a sphere. But how can I find the radius of the inverted sphere?
In other words, let an empty sphere $K=(o,R)$, center $o$ and radius R, and another sphere sitting inside of $K$, $S=(c,r)$. For simplicity, suppose S is not intersecting K, and does not enclose its center $o$ either. Now, under inversion the sphere S will be mapped to another sphere outside the boundary of K, but I also want to know the radius of that sphere, and textbooks like Visual Complex Analysis don't discuss the radius of that sphere, does anybody have any ideas?
NOTE: By inversion I mean the mapping of a point $p$ inside a sphere $Q=(o,R)$, to a point colinear to $o - p$ whose distance from the origin $o$ is $\frac{R^2}{r}$, if the distance of $p$ to the origin is $r$.
The key thing is that the center of the sphere’s inversion is not in general the same as the inversion of the sphere’s center, so you have to work with the diameter instead. The sphere being inverted intersects the line joining the centers at distances $c\pm r$. The inverted diameter length is then just the difference between the images of these distances, which gives for the inverted radius $$ r'=\frac12\left|{R^2 \over c-r}-{R^2 \over c+r}\right| = \left|{R^2 \over c^2-r^2}\right|r.$$ Note that most of the simplifying assumptions you made were unnecessary. This relation holds regardless of how the spheres overlap, including the degenerate case $r=c$.