Preservation of the Inverse Points Property by Extended Mobius Transformation
Mobius transformations map generalized circles (circles and lines) to generalized circles, but they do not map the centres of circles to the centres of circles. They do however, preserve the inverse property of points, so that if
and
are inverse points with respect to a circle
then after being transformed by an extended mobius transformation
and
are inverse points with respect to
Proof
Ifandare inverse points with respect to a generalized circle
then there must be an extended mobius transformation
that mapsto 0,to
andto the unit circle. Thus
mapsto 0,toand
to the unit circle. It follows thatandare inverse points with respect to
On the other hand if
inthen
so once againandare inverse points with respect to