Editing Hyperbolic angle

{{short description|Argument of the hyperbolic functions}}
[[Image:Hyperbolic sector.svg|thumb|200px|right|The curve represents ''xy'' = 1. A hyperbolic angle has magnitude equal to the area  of the corresponding [[hyperbolic sector]], which is in ''standard position'' if {{math|''a'' {{=}} 1}}]]

In [[geometry]], '''hyperbolic angle''' is a [[real number]] determined by the [[area]] of the corresponding [[hyperbolic sector]] of ''xy'' = 1 in Quadrant I of the [[Cartesian plane]]. The hyperbolic angle parametrizes the [[unit hyperbola]], which has [[hyperbolic functions]] as coordinates. In mathematics, hyperbolic angle is an [[invariant measure]] as it is preserved under [[hyperbolic rotation]].

The hyperbola ''xy'' = 1 is [[rectangular hyperbola|rectangular]] with semi-major axis <math>\sqrt 2</math>, analogous to the circular [[angle]] equaling the area of a [[circular sector]] in a circle with radius <math>\sqrt 2</math>.

Hyperbolic angle is used as the [[dependent and independent variables|independent variable]] for the [[hyperbolic functions]] sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular (trigonometric) functions by regarding a hyperbolic angle as defining a [[hyperbolic sector#Hyperbolic triangle|hyperbolic triangle]].
The parameter thus becomes one of the most useful in the [[calculus]] of [[real number|real]] variables.

==Definition==
Consider the rectangular hyperbola <math>\textstyle\{(x,\frac 1 x): x>0\}</math>, and (by convention) pay particular attention to the ''branch'' <math>x > 1</math>.

First define:
* The hyperbolic angle in ''standard position'' is the [[angle]] at <math>(0, 0)</math> between the ray to <math>(1, 1)</math> and the ray to <math>\textstyle(x, \frac 1 x)</math>, where <math>x > 1</math>.
* The magnitude of this angle is the [[area]] of the corresponding [[hyperbolic sector]], which turns out to be <math>\operatorname{ln}x</math>.

Note that, because of the role played by the [[natural logarithm]]:
* Unlike circular angle, the hyperbolic angle is ''unbounded'' (because <math>\operatorname{ln}x</math> is unbounded); this is related to the fact that the [[harmonic series (mathematics)|harmonic series]] is unbounded.
* The formula for the magnitude of the angle suggests that, for <math>0 < x < 1</math>, the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is ''directed''.

Finally, extend the definition of ''hyperbolic angle'' to that subtended by any interval on the hyperbola. Suppose <math>a, b, c, d</math> are [[positive real numbers]] such that <math>ab = cd = 1</math> and <math>c > a > 1</math>, so that <math>(a, b)</math> and <math>(c, d)</math> are points on the hyperbola <math>xy=1</math> and determine an interval on it. Then the [[squeeze mapping]] <math>\textstyle f:(x, y)\to(bx, ay)</math> maps the angle <math>\angle\!\left ((a, b), (0,0), (c, d)\right)</math> to the ''standard position'' angle <math>\angle\!\left ((1, 1), (0,0), (bc, ad)\right)</math>. By the result of [[Gregoire de Saint-Vincent]], the hyperbolic sectors determined by these angles have the same area, which is taken to be the magnitude of the angle. This magnitude is <math>\operatorname{ln}{(bc)}=\operatorname{ln}(c/a) =\operatorname{ln}c-\operatorname{ln}a</math>.

==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.

==History==
The [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] is the evaluation of the area of a [[hyperbolic sector]]. It can be shown to be equal to the corresponding area against an [[asymptote]]. The quadrature was first accomplished by [[Gregoire de Saint-Vincent]] in 1647 in ''Opus geometricum quadrature circuli et sectionum coni''. As expressed by a historian,
: [He made the] quadrature of a hyperbola to its [[asymptote]]s, and showed that as the [[area]] increased in [[arithmetic series]] the [[abscissa]]s increased in [[geometric series]].<ref>[[David Eugene Smith]] (1925) ''History of Mathematics'', pp. 424,5  v. 1</ref>

[[A. A. de Sarasa]] interpreted the quadrature as a [[logarithm]] and thus the geometrically defined [[natural logarithm]] (or "hyperbolic logarithm") is understood as the area under {{nowrap|1=''y'' = 1/''x''}} to the right of {{nowrap|1=''x'' = 1}}.  As an example of a [[transcendental function]], the logarithm is more familiar than its motivator, the hyperbolic angle. Nevertheless, the hyperbolic angle plays a role when the [[Squeeze mapping#Bridge to transcendentals|theorem of Saint-Vincent]] is advanced with [[squeeze mapping]].

Circular [[trigonometry]] was extended to the hyperbola by [[Augustus De Morgan]] in his [[textbook]] ''Trigonometry and Double Algebra''.<ref>[[Augustus De Morgan]] (1849) [https://books.google.com/books?id=7UwEAAAAQAAJ Trigonometry and Double Algebra], Chapter VI: "On the connection of common and hyperbolic trigonometry"</ref> In 1878 [[William Kingdon Clifford|W.K. Clifford]] used the hyperbolic angle to [[parametric equation|parametrize]] a [[unit hyperbola]], describing it as "quasi-[[harmonic oscillator|harmonic motion]]".

In 1894 [[Alexander Macfarlane]] circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generate [[versor#Hyperbolic versor|hyperbolic versors]], in his book ''Papers on Space Analysis''.<ref>[[Alexander Macfarlane]](1894) [https://archive.org/details/principlesalgeb01macfgoog ''Papers on Space Analysis''], B. Westerman, New York</ref> The following year [[Bulletin of the American Mathematical Society]] published [[Mellen W. Haskell]]'s outline of the [[hyperbolic function]]s.<ref>[[Mellen W. Haskell]] (1895) [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9</ref>

When [[Ludwik Silberstein]] penned his popular 1914 textbook on the new [[theory of relativity]], he used the [[rapidity]] concept based on hyperbolic angle ''a'', where {{nowrap|1=[[tanh]] ''a'' = ''v''/''c''}}, the ratio of velocity ''v'' to the [[speed of light]]. He wrote:

:It seems worth mentioning that to ''unit'' rapidity corresponds a huge velocity, amounting to 3/4 of the velocity of light; more accurately we have {{nowrap|1=''v'' = (.7616)''c''}} for {{nowrap|1=''a'' = 1}}.
:[...] the rapidity {{nowrap|1=''a'' = 1}}, [...] consequently will represent the velocity .76&nbsp;''c'' which is a little above the velocity of light in water.

Silberstein also uses [[Nikolai Lobachevsky|Lobachevsky]]'s concept of [[angle of parallelism]] Π(''a'') to obtain {{nowrap|1=cos Π(''a'') = ''v''/''c''}}.<ref>[[Ludwik Silberstein]] (1914) [https://archive.org/details/theoryofrelativi00silbrich The Theory of Relativity], pp.&nbsp;180–1 via [[Internet Archive]]</ref>

==Imaginary circular angle==
The hyperbolic angle is often presented as if it were an [[imaginary number]], <math display=inline> \cos ix = \cosh x</math> and <math display=inline>\sin ix = i \sinh x,</math> so that the [[hyperbolic function]]s cosh and sinh can be presented through the circular functions. But in the Euclidean plane we might alternately consider circular angle measures to be imaginary and hyperbolic angle measures to be real scalars, <math display=inline> \cosh ix = \cos x</math> and <math display=inline>\sinh ix = i \sin x.</math>

These relationships can be understood in terms of the [[exponential function]], which for a complex argument <math display=inline>z</math> can be broken into [[Even and odd functions|even and odd parts]] <math display=inline>\cosh z = \tfrac12(e^z + e^{-z})</math> and <math display=inline>\sinh z = \tfrac12(e^z - e^{-z}),</math> respectively. Then

<math display=block>e^z = \cosh z + \sinh z = \cos(iz) - i \sin(iz), </math>

or if the argument is separated into real and imaginary parts <math display=inline>z = x + iy,</math> the exponential can be split into the product of scaling <math display=inline>e^{x}</math> and rotation <math display=inline>e^{iy},</math>

<math display=block>e^{x + iy} = e^{x}e^{iy} = (\cosh x + \sinh x)(\cos y + i \sin y).</math>

As [[infinite series]],

<math display=block>\begin{alignat}{3}
e^z      &= \,\,\sum_{k=0}^\infty \frac{z^k}{k!}  &&
          = 1 + z + \tfrac{1}{2}z^2 + \tfrac16z^3 + \tfrac1{24}z^4 + \dots \\
\cosh z  &= \sum_{k \text{ even} } \frac{z^k}{k!} &&
          = 1 + \tfrac{1}{2}z^2 + \tfrac1{24}z^4 + \dots \\
\sinh z  &= \,\sum_{k \text{ odd} } \frac{z^k}{k!} &&
          = z + \tfrac{1}{6}z^3 + \tfrac1{120}z^5 + \dots \\
\cos z   &= \sum_{k \text{ even} } \frac{(iz)^k}{k!} &&
          = 1 - \tfrac{1}{2}z^2 + \tfrac1{24}z^4 - \dots \\
i \sin z &= \,\sum_{k \text{ odd} } \frac{(iz)^k}{k!} &&
          = i\left(z - \tfrac{1}{6}z^3 + \tfrac1{120}z^5 - \dots\right) \\
\end{alignat}</math>

The infinite series for cosine is derived from cosh by turning it into an [[alternating series]], and the series for sine comes from making sinh into an alternating series.

==See also==
*[[Transcendent angle]]

==Notes==
{{Reflist}}

==References==
{{wikibooks|Calculus|Hyperbolic angle|Hyperbolic angle}}
* [[Janet Barnett|Janet Heine Barnett]] (2004) "Enter, stage center: the early drama of the hyperbolic functions", available in (a) [[Mathematics Magazine]] 77(1):15–30 or (b) chapter 7 of ''Euler at 300'', RE Bradley, LA D'Antonio, CE Sandifer editors, [[Mathematical Association of America]] {{ISBN|0-88385-565-8}} .
* [[Arthur Kennelly]] (1912) [https://archive.org/details/applicationofhyp00kennrich Application of hyperbolic functions to electrical engineering problems]
* William Mueller, ''Exploring Precalculus'', § The Number e, [http://www.wmueller.com/precalculus/e/e5.html Hyperbolic Trigonometry].
* [[John Stillwell]] (1998) ''Numbers and Geometry'' exercise 9.5.3, p.&nbsp;298, Springer-Verlag {{ISBN|0-387-98289-2}}.

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[[Category:Angle]]
[[Category:Differential calculus]]
[[Category:Integral calculus]]