Hyperbolic motion

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For hyperbolic motion in physics, see hyperbolic motion (relativity).

In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in the model. Such a mapping is analogous to congruences of Euclidean geometry which are compositions of rotations and translations. One uses hyperbolic motions to relate structures within the model. The collection of all hyperbolic motions form a group which characterizes the geometry according to the Erlangen program. Hyperbolic motions are most easily visualized in the upper half-plane model HP = {(x,y): y > 0}.

It is also described with polar coordinates as HP = {(r cos a, r sin a): 0 < a < π, r > 0 }.

Let p = (x,y) or p = (r cos a, r sin a), p ∈ HP.

There are three fundamental hyperbolic motions:

p → q = (x + c, y ), c ∈ R (left or right shift)

p → q = (sx, sy ), s > 0 (dilation)

p → q = ( r⁻¹ cos a, r ⁻¹ sin a ) (reflection in unit semicircle).

The general hyperbolic motion is a composition of fundamental hyperbolic motions. (Compare with Möbius transformations and inversive geometry.)

[edit] Elementary half-plane geometry

Consider the triangle {(0,0),(1,0),(1,tan a)}. Since 1 + tan²a = sec²a, the length of the triangle hypotenuse is sec a (see secant). Set r = sec a and apply the third fundamental hyperbolic motion to obtain q = (r cos a, r sin a) where r = sec⁻¹a = cos a. Now

|q – ( 1/2,0)|² = (cos²a – ½)² +cos²a sin²a = ¼

so that q lies on the semicircle Z of radius ½ and center (1/2,0). Thus the tangent ray at (1,0) gets mapped to Z by the third fundamental hyperbolic motion. Any semicircle can be re-sized by a dilation to radius ½ and shifted to Z, then the reflection carries it to the tangent ray. So the collection of hyperbolic motions permutes the semicircles with diameters on y = 0 sometimes with vertical rays, and vice versa.

Suppose one agrees to measure length on vertical rays by the logarithm function:

d((x,y),(x,z)) = log(z/y).

Then by means of hyperbolic motions one can measure distances between points on semicircles too: first move the points to Z with appropriate shift and dilation, then place them by reflection on the tangent ray where the logarithmic distance is known.

For m and n in HP, let b be the perpendicular bisector of the line segment connecting m and n. If b is parallel to the abscissa, then m and n are connected by a vertical ray, otherwise b intersects the abscissa so there is a semicircle centered at this intersection that passes through m and n. The set HP becomes a metric space when equipped with the distance d(m,n) for m,n ∈ HP as found on the vertical ray or semicircle. One calls the vertical rays and semicircles the hyperbolic lines in HP. Since the erection of the HP model relies deeply on Euclidean geometry and traditional trigonometry (especially tangent and secant), it is natural to consider hyperbolic geometry as meta-Euclidean, not non-Euclidean.

[edit] Disk model motions

Consider the disk D = {z ∈ C : z z* < 1 } in the complex plane C. The hyperbolic plane of Lobachevsky can be displayed in D with circular arcs perpendicular to the boundary of D signifying hyperbolic lines. Suppose a and b are complex numbers with a a* − b b * = 1. Note that

|bz + a *|² − |az + b *|² = (aa * − bb *)(1 − |z|²)

so that |z| < 1 implies |(az + b *)/(bz + a *)| < 1 . Hence the Möbius transformation

f(z) = (az + b *)/(bz + a *)

leaves the disk D invariant.Since it also permutes the hyperbolic lines we see that these transformations are motions of the D model of hyperbolic geometry. A complex matrix

with aa* − bb* = 1, which represents the Möbius transformation from the projective viewpoint, can be considered to be on the unit sphere in the ring of coquaternions.