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 theproportion  $PO:TO=TO:P^{\prime }O$  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 thecomplex plane.  Moreover, the mapping $P\mapsto P^{\prime }$ is always aMöbius 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.