Circle Inversions with Software

Background

Inversion of points

The reflection of a point P across a line m is the point P' for which m is the perpendicular bisector of the segment PP'.

Circle inversion is analogous to reflection over a line, with the circle taking the place of the reflecting line and the distance relationship that the radius of the circle is the geometric mean of the distances of P and P' to the center of the circle.

Denoting the center by Q and the radius by r, the distance relationship is

PQ/r = r/P'Q that is, PQ*P'Q = r2

Specifically, given a circle C with center Q and radius r, the inverse in C of P is the point P' determined by two properties:

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

(ii) PQ*P'Q = r2.

The nonlinearity of inversion necessitates a more careful formulation in coordinates.

Denote the coordinates of the relevant points by (a,b) for Q, (x,y) for P, and (x',y') for P'. Then the distance relation PQ*P'Q = r2 becomes

Solving for (x',y') gives

This is the formula for the inversion of P over C:

(x',y') = P' = IC(P) = IC(x,y)

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