Inversions with respect to an inverse circle
Inversions are
applications defined in the plane except one point.
Given a circle k with radius r and center O and a point A different from the center of the
circle, we define inversion with
respect to k from A
as the point A' which lies in
the ray OA satisfying OA.OA = r2.
We call A' the inverse of A with respect
to k.
We call O the center of the inversion.
Some properties of the inversions:
- If A
is an exterior point (resp. interior) of the circle, then A' is an
interior point (resp. exterior).
- The points
on the circle k are
invariants with respect to the inverse circle k.
- If we apply
a inversion twice we obtain the identity transformation. We say that
inversions are involutive applications.
- Transform
lines which do not contain the center of the inversion to circles which
contain it.
- Transform
circles which contain the center of the inversion to lines which do not
contain it.
- Transform
circles which do not contain the center of the inversion to other
circles which do not contain it either.
- Preserve
the angles, they are conformal applications.
In Hyperbolic
Workshop.pdf
(in Catalan, pages 21-28) the proof of these properties is given
together with some other properties. It is also explained how the
inversion point can be found.
In inversions.gsp
you can find a macro that allows to plot the inverse point with respect
to a circle. To construct this macro were necessary to distinguish two
cases since the point can be in the interior or in the exterior of the
circle.
Relation with the transformations of
the hyperbolic plane:
In the Half-Plane Model, the transformations are composition of
inversions with respect to hyperbolic lines. These hyperbolic lines are
euclidean semicircles with center in the boundary line. From the
properties listed below it can be proved that these inversions are
isometries of the Half-Plane Model of the hyperbolic geometry, that's
to say, they preserve distances and angles. They preserve the angles
since they preserve the euclidean angles and the Half-Plane Model is
conformal with the Euclidean Plane.
Hyperbolic geometry
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