The extended plane and i-lines.

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.

Let E be the euclidean plane. The extended plane E+ consists of
the points of E, together with the point at infinity, ∞

Then, for circle D with centre Q, we extend iD to E+ by defining
iD(Q) = ∞, and iD(∞) = Q.

The extended map (which still has order 2) is also denoted by iD,
and is still called inversion in 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 iL.

If L is a line on E, then we define the extended line L+ to be the line L together with ∞.

For P in E, P is on L
if and only if iL(P)=P.
Since ∞ also has this property,
we regard it as "on" L

In Apollonian Families, we discovered that each apollonian curve was
either a circle or a line. We introduce a term to include both types in the extended plane.

A subset of E+ is an i-line if it is a circle or an extended line.

Basic facts
Suppose that M and N are i-lines. Then

  • M is an extended line if and only if ∞ lies on M.
  • The map iM is defined on E+, and is of order 2.
  • A point P in E+ lies on M if and only if iM(P) = P.
  • If M and N are extended lines, they meet at ∞ (and possibly once more).
  • M and N may meet in up to two points - see the CabriJava pane.
  • Any three distinct points in E+ determine a unique i-line.
Only the last of these needs much justification.
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 i-lines are the "lines" of inversive geometry, and the inversions are the fundamental transformations.

Main inversive page