The inverses of (extended) lines and circles
To make the statement of the following theorem as simple as possible,
we use the notions of the extended lines.
|The Inversion Theorem
Let C be a circle with centre O.
|Proof of the Inversion Theorem||There is a neater (algebraic) proof
Algebraic Inversion Theorem page
CabriJava illustrations of cases 2 and 4
This shows the case where we are inverting
an extended line L in the Circle C.
If you move Q (on L),
you will see that the image always lies on the circle L'.
This shows the case where we are invering
a circle D in the circle C.
If you move P on D,
you will see that P' always lies on the circle D'.
The importance of this result is that it shows that, if J is either a circle or an extended line,
then the inverse will be a circle or extended line.
Thus, the Theorem says that inversion maps i-lines to i-lines!
These i-lines play much the same role in inversive geometry thet lines do in euclidean geometry.
We now look at the the effect of inversion on angles between i-lines.
|Main Inversive Page|