TITLE: Seeking conceptual explanation of these nice bijections on roots of unity
QUESTION [10 upvotes]: I proved the following facts by unenlightening calculations.  Since the statements are quite clean, I think there should be a conceptual explanation for them, which my proof certainly is not.
Let $q$ be a prime power, and let $\mu_{q+1}$ be the set of $(q+1)$-th roots of unity in the finite field $\mathbf{F}_{q^2}$.  If $b\in\mu_{q+1}$ and $c\in\mathbf{F}_{q^2}\setminus\mathbf{F}_q$ then
$$
x\mapsto \frac{cx-bc^q}{x-b}
$$
maps $\mu_{q+1}$ to $\mathbf{F}_q\cup\{\infty\}$.  If $b\in\mu_{q+1}$ and $d\in\mathbf{F}_{q^2}\setminus\mu_{q+1}$ then
$$
x\mapsto \frac{x-bd^q}{dx-b}
$$
maps $\mu_{q+1}$ to itself.  (It is also true that these are the only degree-one rational functions which map $\mu_{q+1}$ to either $\mathbf{F}_q\cup\{\infty\}$ or $\mu_{q+1}$, but I'm mainly interested in understanding the existence.)
I tagged this "group theory" because the first fact vaguely feels like a connection between orbits of a nonsplit torus and a split torus in $\textrm{PGL}_2(q)$.  It's tempting to identify $\mathbf{F}_{q^2}$ with $\mathbf{F}_q\times\mathbf{F}_q$, and consider the resulting action of $\textrm{GL}_2(q)$ on $\mathbf{F}_{q^2}$, but I don't see how to go further in this way.
Any suggestions?

REPLY [9 votes]: The question and the analog to the Cayley map in complex numbers pointed out in the comment by Jyrki Lahtonen is not only an analog, but in fact both are special cases of this more general observation: Let $z\mapsto\bar z$ be an involutory automorphism of a field $F$, and let $E$ be the subfield fixed by $\bar{\phantom{a}}$. Furthermore, set $S=\{z\in F\;|\;z\bar z=1\}$. Then, for $b\in S$, $c\in F\setminus E$, and $d\in F\setminus S$,
\begin{equation*}
z\mapsto \frac{cz-b\bar c}{z-b}
\end{equation*}
maps $S$ to $E\cup\{\infty\}$ and
\begin{equation*}
z\mapsto \frac{z-b\bar d}{dz-b}
\end{equation*}
maps $S$ to $S$.