Inversion and Stereographic Projection

An inversion with respect to a circle in \({\mathbb {R}}^2\) (or with respect to a sphere in \({\mathbb {R}}^3\) ) is a one-to-one transformation of \({\mathbb {R}}^2\) (or \({\mathbb {R}}^3\) ) that interchanges the inside of the circle (sphere) and the outside of the circle (sphere). Inversions have nice properties, and they turn out to be convenient when we study arrangements of circles, or spheres.