inversion of plane

Let c be a fixed circle in the Euclidean planeMathworldPlanetmath with center O and radius r. Set for any point PO of the plane a corresponding point P, called the inverse pointMathworldPlanetmath of P with respect to c, on the closed ray from O through P such that the productPlanetmathPlanetmath


has the value r2. This mappingPP  of the plane interchanges the inside and outside of the base circle c. The point O is the “infinitely distant point” of the plane.

The following is an illustration of how to obtain P for a given circle c and point P outside of c. The restricted tangent from P to c is drawn in blue, the line segmentMathworldPlanetmath that determines P (perpendicularPlanetmathPlanetmathPlanetmath to OP¯, having an endpoint on OP¯, and having its other endpoint at the point of tangency T of the circle and the tangent line) is drawn in red, and the radius OT¯ is drawn in green.


The picture justifies the correctness of P, since the trianglesMathworldPlanetmath OPT and OTP are similarMathworldPlanetmathPlanetmath, implying the proportion  PO:TO=TO:PO  whence  POPO=(TO)2=r2.  Note that this same holds if P and P were swapped in the picture.

Inversion formulae.  If O is chosen as the origin of 2 and  P=(x,y)  and  P=(x,y),  then

x=rxx2+y2,y=ryx2+y2;x=rxx 2+y 2,y=ryx 2+y 2.

Note.  Determining inverse points can also be done in the complex plane.  Moreover, the mapping PP is always a Möbius transformationMathworldPlanetmathPlanetmath.  For example, if  c={z|z|=1},  i.e. (  O=0  and  r=1, then the mapping  PP  is given by f:{}{}  defined by  f(z)=1z.

Properties of inversionMathworldPlanetmath

  • The inversion is involutory, i.e. if  PP,  then  PP.

  • The inversion is inversely conformal, i.e. the intersectionMathworldPlanetmath angle of two curves is preserved (check the Cauchy–Riemann equations (!).

  • A line through the center O is mapped onto itself.

  • Any other line is mapped onto a circle that passes through the center O.

  • Any circle through the center O is mapped onto a line; if the circle intersects the base circle c, then the line passes through both intersection points.

  • Any other circle is mapped onto its homotheticMathworldPlanetmath circle with O as the homothety center.


  • 1 E. J. Nyström: Korkeamman geometrian alkeet sovellutuksineen.  Kustannusosakeyhtiö Otava, Helsinki (1948).
Title inversion of plane
Canonical name InversionOfPlane
Date of creation 2015-06-14 18:40:35
Last modified on 2015-06-14 18:40:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Topic
Classification msc 51K99
Classification msc 53A30
Classification msc 30E20
Synonym mirroring in circle
Synonym circle inversion
Related topic MobiusTransformation
Related topic PowerOfPoint
Defines inverse point
Defines inversion
Defines inversion formulae
Defines involutory