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 non-negative real number.

Suppose $T$ denotes a hyperbolic rotation such that $T(H)\subseteq H$ . Set

$\begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}$

where $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ 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

$\begin{pmatrix}a&0\\ 0&a^{-1}\end{pmatrix}$

Since the matrix is non-singular, 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

$P\begin{pmatrix}a&0\\ 0&a^{-1}\end{pmatrix}P^{-1}$

for some $0\neq a\in\mathbb{R}$ and some orthogonal matrix $P$ . In other words, $T$ is diagonalizable with $a$ and $a^{-1}$ as eigenvalues ( $T$ is non-singular as a result).

Below are some simple properties:

•

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 $H^{\prime}\in[H]$ .
•

$[\cdot]$ defined above partitions $P$ into disjoint subsets. Call each of these subset a sub-pencil. Let $A$ be a sub-pencil of $P$ . Call $T$ fixes $A$ if $T$ fixes any element of $A$ . Let $A\neq B$ be sub-pencils of $P$ . Then $T$ fixes $A$ iff $T$ does not fix $B$ .
•

Let $A, B$ be sub-pencils 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 sub-pencil 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 two-dimensional 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	2013-03-22 17:24:34
Last modified on	2013-03-22 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 51-00
Related topic	Hyperbola2