Inversion

Please enable Java for an interactive construction (with Cinderella).

Probably the most interesting of the mappings discussed here is the Inversion in a circle. For a circle with center

and radius

we define the inversion of a point

by the equation

$|OA|\cdot|OA'|=r^2.$

Viewed from

, the point

should be in the same direction as

In the applet above you can move the point

around and observe its image

. If you move the point

along the circle or house in the right applet, you can see how the image is generated.

The inversion in a circle has remarkable properties:

lines are mapped to circles (or lines),
circles are mapped to circles (or lines).

If one interprets lines as circles with an infinitely large radius, one can conclude: Circles are mapped to circles!

Furthermore, an inversion preserves the angle of intersection of objects. In the above example, one can also move the circles and the house, in order to understand the consequences of the inversion. A few special cases:

Circles through the center of the circle of inversion are mapped to lines
Circles perpendicular to the circle of inversion are mapped to themselves
The inversion exchanges interior and outerior of the circle of inversion
The center of the circle of inversion is taken "to infinity".

Download files:

Reflection $\hookleftarrow$ Contents $\hookrightarrow$ Complex addition and multiplication (integral)?