Inversion

Inversion transformation

This kind of transformation is defined by giving its circle of fixed points or inversion circle (c). For every point P, other than the center O of the circle, define P' on line OP, such that OP*OP'=r², r being the radius of the circle.

[1] The inversion transformation P'=F(P) is an involution i.e. F²=1.
[2] The points of the inversion circle c(O,r) remain fixed by F.
[3] {P,P'} are harmonic conjugate with respect to {X,X'}, later being the intersections of PP' with the inversion circle.
[4] Every circle through {P,P'} intersects the inversion circle c orthogonally and remains invariant by F.
[5] The image under F of a circle (d), not passing through O, is a circle (d').
[6] The image under F of a circle (d), passing through O, is a line and inversely, the image of a line is a circle passing through O.
[7] The inversion is a conformal map i.e. preserves the angle of tangents of two intersecting curves.
[8] The inversion, considered as a map of the complex plane onto itself, preserves the cross-ratio up to conjugacy:

[9] In particular, it preserves the cross ratio of four points on a circle or a line (since later is then real).

Circles (d) intersecting the iversion-circle (c) orthogonaly remain invariant by the inversion on (c).

The image under F of a circle (d), not passing through O, is a circle (d'). Here are two circles that F interchanges. They intersect on the inversion-circle (c). Thus, the radical axis of the pair (d,d') is the same with the radical axis of the pair of circles (c,d). This is true for every circle d, not passing through the inversion center O.

Tangents of circles d, d' inverse to each other with respect to a circle (c) at respectively inverse points P, P' intersect on the radical axis of the two circles.

The image under F of a circle (d), passing through O, is a line d' and inversely, the image of a line d' is a circle d passing through O. Through two pairs of inverse points {P,P'} and {Q,Q'} passes one circle, which is orthogonal to the circle of inversion (c).

Inversions are conformal maps. The tangents of two curves at their intersection point P form an angle equal to the angle of the tangents of the image curves u'=F(u), v'=F(v) at the image P'=F(P). Because for every direction (t) at P, you can pass a circle orthogonal to (c) and tangent to (t) at P. This circle maps to itself and the tangent is mapped to (t') symmetric to (t) with respect to the medial line of segment PP'.

The inversion, considered as a map of the complex plane onto itself, preserves the cross-ratio up to conjugacy. The cross ratios (X,Y,Z,W) and (X',Y',Z',W') are conjugate to each other. If one of them is real then so is the other. Thus, four points on a line or circle define a cross ratio preserved by inversions.

Inversion transformation

See Also