Inverse Points and Apollonian Form
An curve expressed asis said to be written in Apollonian form. A curve written in Apollonian form is in fact either a circle or a line – circles and lines together constitute the set of generalized circles, with a line being considered a circle of infinite radius.
Ifthe curve is a line. The line consists of the set of points equidistant fromand %beta .andare mirror images or inverse points of each other in the between them.
We can generalized inverse points to the case
Letbe a generalized circle.andare inverse points with respect toifand lie onandhas the equationor one of the points, say, is infinity andhas the equationfor some
Suppose thatandare distinct inverse points with respect to a generalized circleThen there exists an extended mobius transformationthat mapsto 0,to infinity andto the unit circle. Letbe the point onsatisfying
mapsonto the unit circle so so forand
Ifthis becomesso that
Ifthenso thatso that
Conversely ifhas equationwiththenand we can defineand ifhas equationandthen we can define
In either casemapsto 0,to infinity andto the unit circle since soandare inverse points with respect to