Inversion in a circle

A Euclidean line may be regarded as the ‘limiting case’ of circles of increasing radius. In this sense, it is not surprising that reflection in a line is the analog of a transformation of inversion in a circle. In this section we give a metric definition of inversion in a circle and prove some of the basic properties of inversions; this seems to be the simplest approach, but, in the next section, we give a treatment more in the spirit of Euclidean geometry.

Let c be a circle in the Euclidean plane E with center 0 and radius r. If P is any point of E other than 0, we define P′ to be the unique point on the line OP, on the same side of 0 as P is, such that |0P|. |OP′| = r2. Clearly the map γc: P ↦ P′ is a bijection from E - {0} to E - {0}. As P approaches 0 along any path, that is, as |0P| approaches 0, its image P′ recedes indefinitely, that is, |0P′| increases without bound. This leads us to define the inversive plane as E* = E ∪ {∞}, the result of adjoining a new point ∞ to E, and to extend γc to a bijection from E* to E* by defining 0γ = ∞ and ∞γc = 0.