hyperbolic rotation

Let 𝔼 be the Euclidean planeMathworldPlanetmath equipped with the Cartesian coordinate system. Recall that given a circle C centered at the origin O, one can define an “ordinary” rotationMathworldPlanetmath R to be a linear transformation that takes any point on C to another point on C. In other words, R⁢(C)⊆C.

Similarly, given a rectangular hyperbolaMathworldPlanetmath (the counterpart of a circle) H centered at the origin, we define a hyperbolic rotationMathworldPlanetmath (with respect to H) as a linear transformation T (on 𝔼) such that T⁢(H)⊆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 x⁢y=r, where r is a non-negative real number.

Suppose T denotes a hyperbolic rotation such that T⁢(H)⊆H. Set


where (abcd) is the matrix representation of T, and x⁢y=x′⁢y′=r. Solving for a,b,c,d and we get a⁢d=1 and b=c=0. In other words, with respect to rectangular hyperbolas of the form x⁢y=r, the matrix representation of a hyperbolic rotation looks like


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 x⁢y=r, it is not hard to solve the general case. Since the two asymptotesMathworldPlanetmath of any rectangular hyperbola H are perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath, by an appropriate change of bases (ordinary rotation), H can be transformed into a rectangular hyperbola H′ whose asymptotes are the x and y axes, so that H′ has the algebraic form x⁢y=r. As a result, the matrix representation of a hyperbolic rotation T with respect to H has the form


for some 0≠a∈ℝ and some orthogonal matrixMathworldPlanetmath P. In other words, T is diagonalizable with a and a-1 as eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath (T is non-singular 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 hyperbolaMathworldPlanetmathPlanetmath 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∈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′)=H′ for any H′∈[H].

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    [⋅] defined above partitionsMathworldPlanetmath 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≠B be sub-pencils of P. Then T fixes A iff T does not fix B.

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    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∘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 compositionMathworldPlanetmath. In fact, it is a group.

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    Let T be a hyperbolic rotation fixing the hyperbola x⁢y=r. Then T fixes its branches (connected componentsMathworldPlanetmath) 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∈H. Then T fixes the (measure of) hyperbolic angle between P and Q. In other words, if α is the measure of the hyperbolic angle between P and Q and, by abuse of notation, let T⁢(α) be the measure of the hyperbolic angle between T⁢(P) and T⁢(Q). Then α=T⁢(α).

The definition of a hyperbolic rotation can be generalized into an arbitrary two-dimensional vector spaceMathworldPlanetmath: it is any diagonalizable linear transformation with a pair of eigenvalues a,b such that a⁢b=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