the inversive plane
Inversion in a circle does something awkward to the ordinary plane: it sends the centre of the circle off to nowhere, and there is no point for it to land on. The fix is elegant — add a single new point, called the point at infinity, that all the far-away directions agree to share, and let inversion send the centre there. The ordinary plane together with this one extra point is the inversive plane, the natural stage on which inversions and Mobius transformations behave perfectly.
Concretely, the inversive plane is the usual plane with exactly one point at infinity adjoined — not a whole line of them as in projective geometry, but a single point that closes the plane up into something topologically like a sphere. On this stage two pleasant simplifications happen. First, the catalogue of basic curves collapses: there is no longer any difference in kind between a straight line and a circle, because a line is now simply a circle that passes through the point at infinity. Second, the transformations of conformal geometry — inversions and the Mobius transformations built from them — become bijections with no exceptions, since the point that used to escape to nowhere now has a home. The model that makes this vivid is stereographic projection: place a sphere touching the plane and project from its north pole; every point of the plane corresponds to a point of the sphere, and the north pole itself is exactly the point at infinity. Lines and circles in the plane both correspond to circles on the sphere.
The inversive plane is the home space of conformal and Mobius geometry, just as the projective plane is the home space of projective geometry — each completes the ordinary plane by adding ideal points so that a class of transformations can act without exception. It is essential to keep it distinct from the projective plane: projective geometry adds a whole line of points at infinity, one for each direction, so parallel lines meet there, whereas the inversive plane adds just one point shared by all directions, so it serves angle-preserving maps rather than incidence-preserving ones. They are different completions for different geometries; do not conflate the single inversive point at infinity with the projective line at infinity.
Place a sphere on the plane and project from its north pole onto the plane below (stereographic projection). Every point of the plane corresponds to exactly one point of the sphere; only the north pole has no ordinary point beneath it, and that north pole is the single point at infinity. A straight line in the plane corresponds to a circle on the sphere through the north pole, confirming that lines are just circles through infinity.
Adding one point at infinity turns the plane into a sphere, where lines and circles merge into one family.
The inversive plane adds a single point at infinity (shared by all directions) and suits angle-preserving maps; do not confuse it with the projective plane, which adds a whole line at infinity (one point per direction) and suits incidence.