- Hyperbolic motion
In
geometry , a hyperbolic motion is a mapping of a model ofhyperbolic geometry that preserves the distance measure in the model. Such a mapping is analogous tocongruence s ofEuclidean geometry which are compositions ofrotation s andtranslation s. One uses hyperbolic motions to relate structures within the model. The collection of all hyperbolic motions form a group which characterizes thegeometry according to theErlangen program .Hyperbolic motions are visualized in theupper half-plane model HP = {("x","y"): "y" > 0} with certain geometric transformations. The half-plane is also described withpolar 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" ) (inversion in unit semicircle).The general hyperbolic motion is a composition of fundamental hyperbolic motions.
Elementary half-plane geometry
Consider the triangle {(0,0),(1,0),(1,tan "a")}. Since 1 + tan2"a" = sec2"a", 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−1"a" = cos "a". Now:|"q" – ( 1/2,0)|2 = (cos2"a" – ½)2 +cos2"a" sin2"a" = ¼so 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 inversion 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 thelogarithm 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 inversion 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 ametric 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.Disk model motions
Consider the disk D = {"z" ∈ C : "z z"* < 1 } in the
complex plane C. The geometric plane ofLobachevsky can be displayed in D with circular arcs perpendicular to the boundary of D signifying "hyperbolic lines". Using the arithmetic and geometry of complex numbers, andMobius transformation s, there is thePoincaré disc model of the hyperbolic plane: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 |("a"z + "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 :q = egin{pmatrix} a & b \ b^* & a^* end{pmatrix}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 ofcoquaternion s.References
* A.S. Smogorzhevsky (1982) "Lobachevskian Geometry",
Mir Publishers , Moscow.
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