Questions related to Inversive Geometry and its applications.

Wikipedia says

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (loosely speaking, a circle with infinite radius). Many difficult problems in geometry become much more tractable when an inversion is applied.

The concept of inversion can be generalized to higher dimensional spaces.

$\newcommand{\inv}{\operatorname*{inv}}$ Inversion through the sphere $S(c,r)$ centered at $c$ with radius $r$, is defined as $$ \inv_{S(c,r)}(x)=c+r^2\frac{x-c}{|x-c|^2} $$ That is, the direction from $c$ to $x$ is preserved, but the new distance from $c$ to $x$ is $r^2$ divided by the old distance. Thus, $S(c,r)$ is preserved by $\inv_{S(c,r)}$ and $\inv_{S(c,r)}$ is its own inverse.