Using the Preservation of the Inverse Points Property to Find the Image of a Generalized Circle Under an Extended Mobius Transformation
Extended mobius transformations preserve the inverse points property, so that if
and
are inverse points with respect to a generalized circle
then
and
are inverse points with respect to the image of the circle,
under the mobius transformation.
We can use this inverse points property to find the equation of the image of a circle with a given equation under an extended mobius transformation.
Example: Find the equation of the image of the circlewith equation
under the extended mobius transformation
The circlehas centre
so that
and infinity are inverse points with respect to
Then
and
are inverse points with respect to
The equation of the image ofmust then be
for some
lies onso that
lies on
and
satisfies
The equation of
is
Example: Find the equation of the image of the circlewith equation
under the extended mobius transformation
The circlehas centre
so thatand infinity are inverse points with respect to
Then
andare inverse points with respect to
The equation of the image ofmust then be
lies onso that
lies on
and
satisfies
The equation ofis