Fractals and Circle Inversions

Michael Frame, Barbara Bemis, Colleen Clancy, and Tatiana Cogevina

Inversion across a circle was introduced by Appolonious of Perga. Though Appolonius' definition of inversion was synthetic, as was all geometry at that time, we give an analytic characterization.

Starting with a circle C with center (a, b) and radius r, and given a point (x, y), the point (x', y') is the inverse of (x, y) across the circle C if

  1. Point (x', y') and point (x, y) lie on the same ray from center (a, b), and
  2. The distance from the center to point (x, y) multiplied by the distance from the center to point (x', y') is equal to r2.

The points (x, y) and (x', y') are inverses across the circle C.


Inversion in a circle has several properties easily derived from the definition.

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