Circle Inversion Fractals

Definition of Inversion

Mirror reflection is familiar: the perpendicular of a point P across a mirror m is the point P' for which PQ = P'Q.

Inversion in a circle was introduced by Apollonius of Perga (b 262 BC) in his last book, Plane Loci. Here is his definition.

Given a point P outside the circle C, draw the segment PQ tangent to C. From Q drop the perpendicular to OP intersecting at P'. The point P' is the inverse of P. Conversesly, given a point P' inside the circle C, draw the perpendicular to OP', intersecting C at Q. The tangent to C at Q intersects the line extending OP' at P. The point P is the inverse of P'.

Equivalently, P' is the inverse of P if

(i) P and P' lie on the same ray from O, and

(ii) OP*OP' = OQ2.

Equivalence is proved by showing OP'Q is similar to OQP, and consequently OP'/OQ = OQ/OP.

Here are the formulas for the inverse of a point and the inverse of a circle.

Return to circle inversion fractals.