Editing Hyperbolic angle (section)

==Comparison with circular angle==
[[Image:Hyperbolic functions-2.svg|thumb|200px|right|The unit hyperbola has a sector with an area half of the hyperbolic angle]]

[[File:HyperbolicAnimation.gif|thumb|right|Circular vs. hyperbolic angle]]

A [[unit circle]] <math> x^2 + y^2 = 1 </math> has a [[circular sector]] with an area half of the circular angle in radians. Analogously, a [[unit hyperbola]] <math> x^2 - y^2 = 1 </math> has a [[hyperbolic sector]] with an area half of the hyperbolic angle.

There is also a projective resolution between circular and hyperbolic cases: both curves are [[conic section]]s, and hence are treated as [[projective range]]s in [[projective geometry]]. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows:

Circular angles can be characterized geometrically by the property that if two [[chord (geometry)|chord]]s ''P''{{sub|0}}''P''{{sub|1}} and ''P''{{sub|0}}''P''{{sub|2}} subtend angles ''L''{{sub|1}} and ''L''{{sub|2}} at the centre of a circle, their sum {{nowrap|''L''{{sub|1}} + ''L''{{sub|2}}}} is the angle subtended by a chord ''P''{{sub|0}}''Q'', where ''P''{{sub|0}}''Q'' is required to be parallel to ''P''{{sub|1}}''P''{{sub|2}}.

The same construction can also be applied to the hyperbola.  If ''P''{{sub|0}} is taken to be the point {{nowrap|(1, 1)}}, ''P''{{sub|1}} the point {{nowrap|(''x''{{sub|1}}, 1/''x''{{sub|1}})}}, and ''P''{{sub|2}} the point {{nowrap|(''x''{{sub|2}}, 1/''x''{{sub|2}})}}, then the parallel condition requires that ''Q'' be the point {{nowrap|(''x''{{sub|1}}''x''{{sub|2}}, 1/''x''{{sub|1}}1/''x''{{sub|2}})}}.  It thus makes sense to define the hyperbolic angle from ''P''{{sub|0}} to an arbitrary point on the curve as a logarithmic function of the point's value of ''x''.<ref>Bjørn Felsager, [http://www.dynamicgeometry.com/Documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf Through the Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry] {{Webarchive|url=https://web.archive.org/web/20110716173907/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf|date=2011-07-16}}, ICME-10 Copenhagen 2004; p.14.  See also example sheets [http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_1.pdf] {{Webarchive|url=https://web.archive.org/web/20090106144140/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_1.pdf|date=2009-01-06}} [http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_2.pdf] {{Webarchive|url=https://web.archive.org/web/20081121024206/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_2.pdf|date=2008-11-21}} exploring Minkowskian parallels of some standard Euclidean results</ref><ref>Viktor Prasolov and Yuri Solovyev (1997) ''Elliptic Functions and Elliptic Integrals'', page 1, Translations of Mathematical Monographs volume 170, [[American Mathematical Society]]</ref>

Whereas in [[Euclidean geometry]] moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a [[pseudo-Euclidean space|pseudo-Euclidean plane]] steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the  multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.<ref>[http://www.math.cornell.edu/~web4520/CG15-0.pdf Hyperbolic Geometry] pp 5–6, Fig 15.1</ref>

Both circular and hyperbolic angle provide instances of an [[invariant measure]]. Arcs with an angular magnitude on a circle generate a [[measure (mathematics)|measure]] on certain [[measurable set]]s on the circle whose magnitude does not vary as the circle turns or [[rotation|rotates]]. For the hyperbola the turning is by [[squeeze mapping]], and the hyperbolic angle magnitudes stay the same when the plane is squeezed by a mapping
:(''x'', ''y'') ↦ (''rx'', ''y'' / ''r''), with ''r'' > 0 .

===Relation To The Minkowski Line Element===
There is also a curious relation to a hyperbolic angle and the metric defined on [[Minkowski space]]. Just as two dimensional Euclidean geometry defines its [[line element]] as
:<math>ds_{e}^2 = dx^2 + dy^2,</math>
the line element on Minkowski space is<ref>{{cite web |last1=Weisstein |first1=Eric W. |title=Minkowski Metric |url=https://mathworld.wolfram.com/MinkowskiMetric.html |website=mathworld.wolfram.com |language=en}}</ref>
:<math>ds_{m}^2 = dx^2 - dy^2.</math>

Consider a curve embedded in two dimensional Euclidean space,
:<math>x = f(t), y=g(t).</math>
Where the parameter <math>t</math> is a real number that runs between <math> a </math> and <math> b </math> (<math> a\leqslant t<b </math>). The arclength of this curve in Euclidean space is computed as:
:<math>S = \int_{a}^{b}ds_{e} =  \int_{a}^{b} \sqrt{\left (\frac{dx}{dt}\right )^2 + \left (\frac{dy}{dt}\right )^2 }dt.</math>

If <math> x^2 + y^2 = 1 </math> defines a unit circle, a single parameterized solution set to this equation is <math> x = \cos t </math> and <math> y = \sin t </math>. Letting <math> 0\leqslant t < \theta </math>, computing the arclength <math> S </math> gives <math> S = \theta </math>. Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,
:<math>S = \int_{a}^{b}ds_{m} =  \int_{a}^{b} \sqrt{\left (\frac{dx}{dt}\right )^2 - \left (\frac{dy}{dt}\right )^2 }dt,</math>
and defining a unit hyperbola as <math> y^2 - x^2 = 1 </math> with its corresponding parameterized solution set <math> y = \cosh t </math> and <math> x = \sinh t </math>, and by letting <math> 0\leqslant t < \eta </math> (the hyperbolic angle), we arrive at the result of <math> S = \eta </math>. Just as the circular angle is the length of a circular arc using the Euclidean metric, the hyperbolic angle is the length of a hyperbolic arc using the Minkowski metric.