Let $c$ be a fixed circle in the Euclidean plane with center $O$ and radius $r$ . Set for any point $P \neq O$ of the plane a corresponding point $P'$ , called the inverse point of $P$ with respect to $c$ , on the closed ray from $O$ through $P$ such that the product $$P'O \cdot PO$$ has the constant value $r^2$ . This mapping $P \mapsto P'$ 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 segment that determines $P'$ (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.

Inversion formulae. If $O$ is chosen as the origin of $\mathbb{R}^2$ and $P = (x,\,y)$ and $P' = (x',\,y')$ , then $$x' = \frac{rx}{x^2+y^2},\;\;y' = \frac{ry}{x^2+y^2}; \quad\; x = \frac{rx'}{x'^{\,2}+y'^{\,2}},\;\;y = \frac{ry'}{x'^{\,2}+y'^{\,2}}.$$

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

Properties of inversion

• The inversion is involutory, i.e. if $P\mapsto P'$ , then $P'\mapsto P$ .
• The inversion is inversely conformal, i.e. the intersection 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 homothetic circle with $O$ as the homothety center.

Bibliography

1

E. J. NYSTRÖM: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).