inversion of plane

Let $c$ be a fixed circle in the Euclidean plane with center $O$ and radius $r$. Set for any point $P\ne O$ of the plane a corresponding point $P^{\prime }$, called the inverse point of $P$ with respect to $c$, on the closed ray from $O$ through $P$ such that the product

$$P^{\prime }O\cdot PO$$

has the value $r^2$. This mapping  $P\mapsto P^{\prime }$  of the plane interchanges the inside and outside of the base circle $c$. The point $O^{\prime }$ is the “infinitely distant point” of the plane.

The following is an illustration of how to obtain $P^{\prime }$ for a given circle $c$ and point $P$ outside of $c$. The restricted tangent from $P$ to $c$ is drawn in blue, the line segment that determines $P^{\prime }$ (perpendicular to $\overline{OP}$, having an endpoint on $\overline{OP}$, 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 $\overline{OT}$ is drawn in green.

The picture justifies the correctness of $P^{\prime }$, since the triangles $\mathrm{△}OPT$ and $\mathrm{△}OT{P}^{\prime }$ are similar, implying the proportion  $PO:TO=TO:P^{\prime }O$  whence  $P^{\prime }O\cdot PO={(TO)}^2=r^2$.  Note that this same holds if $P$ and $P^{\prime }$ were swapped in the picture.

Inversion formulae.  If $O$ is chosen as the origin of ${ℝ}^2$ and  $P=(x,y)$  and  $P^{\prime }=(x^{\prime },y^{\prime })$,  then

$$x^{\prime }=\frac{rx}{x^2+y^2},y^{\prime }=\frac{ry}{x^2+y^2};x=\frac{r{x}^{\prime }}{x^{\prime \mathrm{\hspace{0.17em}2}}+y^{\prime \mathrm{\hspace{0.17em}2}}},y=\frac{r{y}^{\prime }}{x^{\prime \mathrm{\hspace{0.17em}2}}+y^{\prime \mathrm{\hspace{0.17em}2}}}.$$

Note.  Determining inverse points can also be done in the complex plane.  Moreover, the mapping $P\mapsto P^{\prime }$ is always a Möbius transformation.  For example, if  $c=\{z\in \mathbb{Z}\mathrm{⋮}|z|=1\}$,  i.e. (http://planetmath.org/Ie)  $O=0$  and  $r=1$, then the mapping  $P\mapsto P^{\prime }$  is given by $f:ℂ\cup \{\mathrm{\infty }\}\to ℂ\cup \{\mathrm{\infty }\}$  defined by  $f(z)={\displaystyle \frac{1}{z}}$.

Properties of inversion

• The inversion is involutory, i.e. if  $P\mapsto P^{\prime }$,  then  $P^{\prime }\mapsto P$.

• The inversion is inversely conformal, i.e. the intersection angle of two curves is preserved (check the Cauchy–Riemann equations (http://planetmath.org/CauchyRiemannEquations)!).

• 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 homothetic circle with $O$ as the homothety center.

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