The extended plane and ilines.
Suppose that C is a circle with centre O. We noted earlier that there is no inverse of O with respect to C, and that there is no point P with inverse O.
Definition
Then, for circle D with centre Q, we extend i_{D} to E^{+} by
defining
The extended map (which still has order 2) is also denoted by i_{D},


Now suppose that L is a line in E.
Then reflection in L is defined for all points of E, so the only way to extend the map to E^{+} is to map ∞ to itself. We refer to the extended map as inversion in L, and denote it by i_{L}. If L is a line on E, then we define the extended line L^{+} to be the line L together with ∞. 
Motivation For P in E, P is on L if and only if i_{L}(P)=P. Since ∞ also has this property, we regard it as "on" L

In Apollonian Families, we discovered that each apollonian curve was
Definition
Basic facts
Suppose that the three points lie in E. Then either they are collinear (so define only a line), or they determine a unique circle (the circumcircle).

Drag A or B to vary the circles, or drag the line. Watch the intersections change. 
These ilines are the "lines" of inversive geometry, and the inversions are the fundamental transformations.
Main inversive page 