14.5: On Inversive Transformations

Recall that the inversive plane is the Euclidean plane with an added point at infinity, denoted by $\infty$. We assume that every line passes thru $\infty$. Recall that the term circline stands for circle or line.

An inversive transformation is a bijection from the inversive plane to itself that sends circlines to circlines. Inversive geometry studies the circline incidence structure of the inversive plane (it sees which points lie on which circlines and nothing else).

Theorem $\PageIndex{1}$

A map from the inversive plane to itself is an inversive transformation if and only if it can be presented as a composition of inversions and reflections.

Exercise 18.8.3 gives another description of inversive transformations by means of complex coordinates.

Proof

Evidently, reflection is an inversive transformation — it maps lines to lines and circles to circles. According to Theorem 10.3.1, any inversion is an inversive transformation as well. Therefore, the same holds for any composition of inversions and reflections.

To prove the converse, fix an inversive transformation $\alpha$.

Assume $\alpha(\infty)=\infty$. Recall that any circline passing thru $\infty$ is a line. If follows that $\alpha$ maps lines to lines; that is, $\alpha$ is an affine transformation that also maps circles to circles.

Note that any motion or scaling (defined in Exercise 14.1.2b) are affine transformations that map circles to circles. Composing $\alpha$ with motions and scalings, we can obtain another affine transformation $\alpha'$ that maps a given unit circle $\Gamma$ to itself. By Exercise 14.2.3, $\alpha'$ fixes the center, say $O$, of the circle $\Gamma$.

Set $P'=\alpha'(P)$. It follows that if $OP=1$, then $OP'=1$. By Proposition 14.3.1, $OP=OP'$ for any point $P$. Finally, by Exercise 14.3.1, we have that if $\overrightarrow{XY} = \overrightarrow{OP}$, then $\overrightarrow{X'Y'}=\overrightarrow{O'P'}$. It follows that $XY=X'Y'$ for any points $X$ and $Y$; that is, $\alpha'$ is a motion.

Summarizing the discussion above, $\alpha$ is a composition of motions and scalings. Observe that any scaling is a composition of two inversions in concentric circles. Recall that any motion is a composition of reflections (see Exercise 5.4.1). Whence $\alpha$ is a composition of inversions and reflections.

In the remaining case $\alpha(\infty) \ne \infty$, set $P=\alpha(\infty)$. Consider an inversion $\beta$ in a circle with center at $P$ and set $\gamma=\beta \circ \alpha$. Note that $\beta(P)=\infty$; therefore, $\gamma(\infty)=\infty$. Since $\alpha$ and $\beta$ are inversive, so is $\gamma$. From above we get that $\gamma$ is a composition of reflections and inversions. Since $\beta$ is self-inverse, we get $\alpha= \beta \circ \gamma$; therefore $\alpha$ is a composition of reflections and inversions as well.

Exercise $\PageIndex{1}$

Show that inversive transformations preserve the angle between arcs up to sign.

More precisely, assume $A'B_1'C_1'$, $A'B_2'C_2'$ are the images of two arcs $AB_1C_1$, $AB_2C_2$ under an inversive transformation. Let $\alpha$ and $\alpha'$ denote the angle between the tangent half-lines to $AB_1C_1$ and $AB_2C_2$ at $A$ and the angle between the tangent half-lines to $A'B_1'C_1'$ and $A'B_2'C_2'$ at $A'$ respectively. Then

$\alpha'=\pm \alpha.$

Hint

Apply Theorem 10.6.1 and Theorem $\PageIndex{1}$

Exercise $\PageIndex{2}$

Show that any reflection can be presented as a composition of three inversions.

Hint

Fix a line $\ell$. Choose a circle $\Gamma$ with its center not on $\ell$. Let $\Omega$ be the inverse of $\ell$ in $\Gamma$; not that $\Omega$ is a circle.

Let $\iota_{\Gamma}$ and $\iota_{\Omega}$ dentoe the inversions in $\Gamma$ and $\Omega$. Apply Corollary 10.6.1 to show that the composition $\iota_{\Gamma} \circ \iota_{\Omega} \circ \iota_{\Gamma}$ is the reflection across $\ell$.

The exercise above implies a stronger version of Theorem $\PageIndex{1}$; namely any inversive transformation is a composition of inversions — no reflections needed.