Definition ICD1
An inversion of a plane \(p\) with respect to a circle \(c\) with a center at \(O\) and a radius \(r\) is a transformation of \(p\) when a point \(A\) is taken into a point \(A'\) in such a way that:
Condition ICC1: \(A\) and \(A'\) are different from \(O\) and \(P_{\infty}\)
Condition ICC2: \(O\), \(A\) and \(A'\) are collinear
Condition ICC3: \(OA \times OA' = r^2\)
The circle \(c\) is called the circle of inversion.
The point \(O\) is called the center of inversion.
The radius \(r\) is called the radius of inversion.
The square of the radius of inversion is called the power of inversion.
The point \(A'\) is also called an image of \(A\) under the inversion with respect to \(c\).
If \(A\) and \(A'\) are both located on one side of \(O\) then the power of inversion is said to be positive, \(+r^2\) or simply \(r^2\).
If \(A\) and \(A'\) are located on either side of \(O\) then the power of inversion is said to be negative, \(-r^2\). The negative sign in front of the power of inversion is largely symbolic - it is used to distinguish between the two possible locations of \(A\) and \(A'\) relative to \(O\) and we will mostly omit it unless required by the context.
As such, we will refer to the power of inversion verbally as in "inversion with negative power" or "inversion with positive power".
If the qualifier of the power of inversion is omitted then it means that the inversion in question is applicable to both power types.
Under the inversion with respect to a circle the center of inversion \(O\) does not really have an image. This follows from ICC3 directly if we diminish the length of the line segment \(OA\) without bound. For completeness, though, we shall take it that the center of inversion \(O\) is transformed into \(P_{\infty}\) and \(P_{\infty}\) is transformed into the center of inversion \(O\).
\(\blacksquare\)