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.
Then, for circle D with centre Q, we extend iD to E+ by
The extended map (which still has order 2) is also denoted by iD,
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
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|