Wilson Stothers' Inversive Geometry and CabriJava Pages

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.
Definition
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 i_D to E⁺ by defining
i_D(Q) = ∞, and i_D(∞) = Q.
The extended map (which still has order 2) is also denoted by i_D,
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 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
either a circle or a line. We introduce a term to include both types in the extended plane.
Definition
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 i_M is defined on E⁺, and is of order 2.
A point P in E⁺ lies on M if and only if i_M(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

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 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 i_D to E⁺ by defining i_D(Q) = ∞, and i_D(∞) = Q. The extended map (which still has order 2) is also denoted by i_D, 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 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 either a circle or a line. We introduce a term to include both types in the extended plane. Definition 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 i_M is defined on E⁺, and is of order 2. A point P in E⁺ lies on M if and only if i_M(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.