Inversion with Respect to a Circle

Louis A. Talman
Associate Professor of Mathematical Sciences
Metropolitan State College of Denver

Inversion with respect to a circle is an important geometric transformation that has applications to a large number of geometric problems. Here we simply give a definition and make a few observations.

Definition: Consider a circle, centered at a point O. Let P and P' be a pair of points in the plane, with P lying inside the circle (but distinct from O) and P' lying outside of the circle. Then P and P' are inverse to each other with respect to the given circle if

Diagram showing construction of inverses
  1. They are colinear with O, and
  2. P lies on OP at the foot of the perpendicular from the point Q where the line connecting P' and Q is tangent to the circle.
Each point of the circle itself is its own inverse. The point O has no inverse.

As far as we are concerned here, the interesting property of inversion with respect to a given circle is this: Given two circles, one centered at O and the other at O', chosen so that the tangent lines to the circles at each point of intersection are perpendicular to each other at that intersection point,

Orthogonal Circles

then inversion about either circle carries the points of the other circle back on to that other circle. Moreover, an inversion about one circle also carries points that lie inside the other circle to other points that lie inside that other circle in one-to-one fashion. Thus, in the picture above, inversion about the circle centered at O interchanges the points P and P' and carries all of the points of the interval PP' back into PP'.

The circle centered at O partitions the disk enclosed by the circle centered at O' into two parts; inversion about the circle centered at O maps each of these parts onto the other in one-to-one fashion. When we consider the open disk inside the circle centered at O' as the hyperbolic plane and represent lines in that plane by the arcs where orthogonal circles intersect that disk, inversions about those orthogonal circles play the same role that reflections about lines play in Euclidean plane geometry.

We will use these observations to transform a mandala by choosing a circle auxiliary to the mandala's bounding circle and orthogonal to that bounding circle. We will then invert the mandala with respect to the auxiliary circle.

This page was constructed on February 8, 2000.
Back to the Mandalas page.