Section 1.8 Reflection across a circle
Recall that for the unit circle \(\T\text{,}\) we have the nice reflection map \(z \mapsto 1/\cc{z}\text{,}\) which maps an element \(z\in \D\) to its reflection across the circle \(\tilde{z}\text{,}\) which lies on the same ray - that is, the map \(z \mapsto \tilde{z}\) sends
\begin{equation*}
r e^{i\theta} \mapsto \frac{1}{r} e^{i\theta}.
\end{equation*}
Notice then that the map \(z \mapsto \tilde{z}\) takes the disk \(\D\) to what we might call the outer disk \(\tilde{\D}\text{.}\)
We're going to extend this idea to general circles (including lines, which should be thought of as circles of infinite radius in this context).
Proof.
In the case where \(C\) is a line, the argument follows from basic geometry, and \(\tilde{z}\) is the point on the line perpendicular to \(C\) passing through \(z\) and equidistant to \(C\) with \(z\text{.}\)
If \(C\) is a circle, then the interior of \(C\) is a disk, and we can conformally map this disk with a LFT \(T\) to the upper half plane, with the circle mapping to the real line (why?). Now the family of circles passing through \(z\) and orthogonal to \(C\) continue to pass through the point \(w = T(z)\) and remain orthogonal to the image of \(C\text{.}\) By the line case, these circles must also pass through the reflection \(\tilde{w}\text{.}\) Finally, we conclude that \(T\inv(\tilde{w}) = \tilde{z}\) is the reflection of \(z\text{.}\)
Proposition 1.8.2. LFTs preserve reflection.
If \(T\) is a linear fractional tranform and \(C\) is a circle (or line), then \(T(\tilde{z}) = \tilde{T}(z)\text{.}\)The next proposition gives a formula for computing \(\tilde{z}\) in the case that \(C\) is a general circle, generalizing the map that sends \(z \to 1/\cc{z}\) for the unit circle. In the case that \(C\) is a line, we can just set \(\tilde{z}\) to be the perpendicular reflection across the line.
Proposition 1.8.3. Reflection formula.
If \(C\) is a circle with center \(z_0\) and radius \(R\text{,}\) then the map \(z \mapsto \tilde{z}\) is a composition of LFTs and complex conjugation. A formula for the reflection is
\begin{equation*}
\tilde{z} = \cc{\frac{\cc{z_0}z + R^2 - \abs{z_0}^2}{z - z_0}}
\end{equation*}
We conclude with the following proposition concerning the properties of the reflection map.
Proposition 1.8.4. Properties of reflection.
- \(\displaystyle \tilde{\tilde{z}} = z\)
- \(z \mapsto \tilde{z}\) preserves the magnitude of angles, but reverses orientation (just as conjugation does).
- \(z \mapsto \tilde{z}\) maps circles to circles.
