Circular Inversion
Note: Page is under construction
Circular Inversion, sometimes called Geometric Inversion, is a transformation where point in the Cartesian plane is transformed based on a circle
with radius
and center
such that
, where
is the transformed point on the ray extending from
through
.
Note that , when inverted, transforms back to
. All points outside of
are transformed inside
, and vice versa. Points on
transform to themselves, meaning
. Finally, the transformation of
is debated on its existence. Some call the transformation the ideal point, which is infinitely far away and in every direction. Others claim that this point does not have an inverse.
Geometric Inversion technically refers to many different types of inversions, however, if Geometric Inversion is used without clarification, Circular Inversion is usually assumed.
Circular Inversion can be a very useful tool in solving problems involving many tangent circles and/or lines.
Basics of Circular Inversion
Inversion of a Circle intersecting O
The first thing that we must learn about inversion is what happens when a circle which intersects the center of the inversion, , is inverted. Let us have circle
, with diameter
.
is chosen arbitrarily on circle
. Points
and
represent the inversions of
and
, respectively.
is the radius of
.
By the definition of inversion, we have and
.
We can combine the two equations to get . Rewriting this gives:
Also, since is a diameter of circle
,
must be right.
Now, we consider and
. They share an angle -
, and we know that
Therefore, we have SAS similarity. Therefore,
must be right. From there, it follows that all points on circle
will be inverted onto the line perpendicular to
at
.
Therefore, the inversion of circle becomes a line.
Note that, if circle extends beyond
, the argument still holds. All one needs to do is shuffle things around.