hyperbolic rotation
Let $\mathbb{E}$ be the Euclidean plane^{} equipped with the Cartesian coordinate system. Recall that given a circle $C$ centered at the origin $O$, one can define an “ordinary” rotation^{} $R$ to be a linear transformation that takes any point on $C$ to another point on $C$. In other words, $R(C)\subseteq C$.
Similarly, given a rectangular hyperbola^{} (the counterpart of a circle) $H$ centered at the origin, we define a hyperbolic rotation^{} (with respect to $H$) as a linear transformation $T$ (on $\mathbb{E}$) such that $T(H)\subseteq H$.
Since a hyperbolic rotation is defined as a linear transformation, let us see what it looks like in matrix form. We start with the simple case when a rectangular hyperbola $H$ has the form $xy=r$, where $r$ is a nonnegative real number.
Suppose $T$ denotes a hyperbolic rotation such that $T(H)\subseteq H$. Set
$\left(\begin{array}{c}\hfill {x}^{\prime}\hfill \\ \hfill {y}^{\prime}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)$
where $\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ is the matrix representation of $T$, and $xy={x}^{\prime}{y}^{\prime}=r$. Solving for $a,b,c,d$ and we get $ad=1$ and $b=c=0$. In other words, with respect to rectangular hyperbolas of the form $xy=r$, the matrix representation of a hyperbolic rotation looks like
$\left(\begin{array}{cc}\hfill a\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {a}^{1}\hfill \end{array}\right)$
Since the matrix is nonsingular, we see that in fact $T(H)=H$.
Now that we know the matrix form of a hyperbolic rotation when the rectangular hyperbolas have the form $xy=r$, it is not hard to solve the general case. Since the two asymptotes^{} of any rectangular hyperbola $H$ are perpendicular^{}, by an appropriate change of bases (ordinary rotation), $H$ can be transformed into a rectangular hyperbola ${H}^{\prime}$ whose asymptotes are the $x$ and $y$ axes, so that ${H}^{\prime}$ has the algebraic form $xy=r$. As a result, the matrix representation of a hyperbolic rotation $T$ with respect to $H$ has the form
$P\left(\begin{array}{cc}\hfill a\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {a}^{1}\hfill \end{array}\right){P}^{1}$
for some $0\ne a\in \mathbb{R}$ and some orthogonal matrix^{} $P$. In other words, $T$ is diagonalizable with $a$ and ${a}^{1}$ as eigenvalues^{} ($T$ is nonsingular as a result).
Below are some simple properties:

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Unlike an ordinary rotation $R$, where $R$ fixes any circle centered at $O$, a hyperbolic rotation $T$ fixing one rectangular hyperbola centered at $O$ may not fix another hyperbola^{} of the same kind (as implied by the discussion above).

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Let $P$ be the pencil of all rectangular hyperbolas centered at $O$. For each $H\in P$, let $[H]$ be the subset of $P$ containing all hyperbolas whose asymptotes are same as the asymptotes for $H$. If a hyperbolic rotation $T$ fixing $H$, then $T({H}^{\prime})={H}^{\prime}$ for any ${H}^{\prime}\in [H]$.

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$[\cdot ]$ defined above partitions^{} $P$ into disjoint subsets. Call each of these subset a subpencil. Let $A$ be a subpencil of $P$. Call $T$ fixes $A$ if $T$ fixes any element of $A$. Let $A\ne B$ be subpencils of $P$. Then $T$ fixes $A$ iff $T$ does not fix $B$.

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Let $A,B$ be subpencils of $P$. Let $T,S$ be hyperbolic rotations such that $T$ fixes $A$ and $S$ fixes $B$. Then $T\circ S$ is a hyperbolic rotation iff $A=B$.

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In other words, the set of all hyperbolic rotations fixing a subpencil is closed under composition^{}. In fact, it is a group.

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Let $T$ be a hyperbolic rotation fixing the hyperbola $xy=r$. Then $T$ fixes its branches (connected components^{}) iff $T$ has positive eigenvalues.

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$T$ preserves area.

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Suppose $T$ fixes the unit hyperbola $H$. Let $P,Q\in H$. Then $T$ fixes the (measure of) hyperbolic angle between $P$ and $Q$. In other words, if $\alpha $ is the measure of the hyperbolic angle between $P$ and $Q$ and, by abuse of notation, let $T(\alpha )$ be the measure of the hyperbolic angle between $T(P)$ and $T(Q)$. Then $\alpha =T(\alpha )$.
The definition of a hyperbolic rotation can be generalized into an arbitrary twodimensional vector space^{}: it is any diagonalizable linear transformation with a pair of eigenvalues $a,b$ such that $ab=1$.
Title  hyperbolic rotation 

Canonical name  HyperbolicRotation 
Date of creation  20130322 17:24:34 
Last modified on  20130322 17:24:34 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 53A04 
Classification  msc 51N20 
Classification  msc 5100 
Related topic  Hyperbola2 