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Hyperbolic angle

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Argument of the hyperbolic functions

The curve representsxy = 1. A hyperbolic angle has magnitude equal to the area of the correspondinghyperbolic sector, which is instandard position ifa = 1

Ingeometry,hyperbolic angle is areal number determined by thearea of the correspondinghyperbolic sector ofxy = 1 in Quadrant I of theCartesian plane. Hyperbolic angle is a shuffled form ofnatural logarithm as they both are defined as an area against hyperbolaxy = 1, and they both are preserved bysqueeze mappings since those mappings preserve area.

The hyperbolaxy = 1 isrectangular with semi-major axis ${\sqrt {2}}$ , analogous to the circularangle equaling the area of acircular sector in a circle with radius ${\sqrt {2}}$ .

Hyperbolic angle is used as theindependent variable for thehyperbolic 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 ahyperbolic triangle. The hyperbolic angle parametrizes theunit hyperbola, which has hyperbolic functions as coordinates.

Definition

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POQ = POS + PQRS − QOR. Equality of areas POS and QOR implies area POQ = area PQRS = $\int _{S}^{R}{\frac {dx}{x}}=\ln R-\ln S=ln{\frac {R}{S}}$ .

{\displaystyle \int _{S}^{R}{\frac {dx}{x}}=\ln R-\ln S=ln{\frac {R}{S}}} — POQ = POS + PQRS − QOR. Equality of areas POS and QOR implies area POQ = area PQRS = $\int _{S}^{R}{\frac {dx}{x}}=\ln R-\ln S=ln{\frac {R}{S}}$ .

Consider the rectangular hyperbola $\textstyle \{(x,{\frac {1}{x}}):x>0\}$ , and (by convention) pay particular attention to the part with $x>1$ .

First define:

The hyperbolic angle instandard position is the angle at $(0,0)$ between the ray to $(1,1)$ and the ray to $\textstyle (x,{\frac {1}{x}})$ , where $x>1$ .
The magnitude of this angle is thesigned area of the correspondinghyperbolic sector, which turns out to be $\operatorname {ln} x$ .

Note that by properties ofnatural logarithm:

Unlike circular angle, the hyperbolic angle isunbounded (because $\operatorname {ln} x$ is unbounded); this is related to the fact that theharmonic series is unbounded.
The formula for the magnitude of the angle suggests that, for $0<x<1$ , the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle isdirected.

Finally, extend the definition ofhyperbolic angle to that subtended by any interval on the hyperbola. Suppose $a, b, c, d {\displaystyle a,b,c,d}$ arepositive real numbers such that $ab=cd=1$ and $c>a>1$ , so that $(a,b)$ and $(c,d)$ are points on the hyperbola $xy=1$ and determine an interval on it. Then thesqueeze mapping $\textstyle f:(x,y)\to (bx,ay)$ maps the angle $\angle \!\left((a,b),(0,0),(c,d)\right)$ to thestandard position angle $\angle \!\left((1,1),(0,0),(bc,ad)\right)$ . By the result ofGregoire 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 $\operatorname {ln} {(bc)}=\operatorname {ln} (c/a)=\operatorname {ln} c-\operatorname {ln} a$ .

Comparison with circular angle

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The unit hyperbola has a sector with an area half of the hyperbolic angle

Aunit circle $x^{2}+y^{2}=1$ has acircular sector with an area half of the circular angle in radians. Analogously, aunit hyperbola $x^{2}-y^{2}=1$ has ahyperbolic sector with an area half of the hyperbolic angle.

There is also a projective resolution between circular and hyperbolic cases: both curves areconic sections, and hence are treated asprojective ranges inprojective 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 twochordsP₀P₁ andP₀P₂ subtend anglesL₁ andL₂ at the centre of a circle, their sumL₁ +L₂ is the angle subtended by a chordP₀Q, whereP₀Q is required to be parallel toP₁P₂.

The same construction can also be applied to the hyperbola. IfP₀ is taken to be the point(1, 1),P₁ the point(x₁, 1/x₁), andP₂ the point(x₂, 1/x₂), then the parallel condition requires thatQ be the point(x₁x₂, 1/x₁1/x₂). It thus makes sense to define the hyperbolic angle fromP₀ to an arbitrary point on the curve as a logarithmic function of the point's value ofx.^[1]^[2]

Whereas inEuclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in apseudo-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.^[3]

Both circular and hyperbolic angle provide instances of aninvariant measure. Arcs with an angular magnitude on a circle generate ameasure on certainmeasurable sets on the circle whose magnitude does not vary as the circle turns orrotates. For the hyperbola the turning is bysqueeze mapping, and the hyperbolic angle magnitudes stay the same when the plane is squeezed by a mapping

(x,y) ↦ (rx,y /r), withr > 0 .

Relation To The Minkowski Line Element

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There is also a relation to a hyperbolic angle and the metric defined onMinkowski space. Just as two dimensional Euclidean geometry defines itsline element as

ds_{e}^{2}=dx^{2}+dy^{2},

the line element on Minkowski space is^[4]

ds_{m}^{2}=dx^{2}-dy^{2}.

Consider a curve embedded in two dimensional Euclidean space,

x=f(t),y=g(t).

Where the parameter $t {\displaystyle t}$ is a real number that runs between $a {\displaystyle a}$ and $b {\displaystyle b}$ ( $a\leqslant t<b$ ). The arclength of this curve in Euclidean space is computed as:

S=\int _{a}^{b}ds_{e}=\int _{a}^{b}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt.

If $x^{2}+y^{2}=1$ defines a unit circle, a single parameterized solution set to this equation is $x=\cos t$ and $y=\sin t$ . Letting $0\leqslant t<\theta$ , computing the arclength $S {\displaystyle S}$ gives $S=\theta$ . Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,

S=\int _{a}^{b}ds_{m}=\int _{a}^{b}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}-\left({\frac {dy}{dt}}\right)^{2}}}dt,

and defining a unit hyperbola as $y^{2}-x^{2}=1$ with its corresponding parameterized solution set $y=\cosh t$ and $x=\sinh t$ , and by letting $0\leqslant t<\eta$ (the hyperbolic angle), we arrive at the result of $S=\eta$ . 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

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Thequadrature of thehyperbola is the evaluation of the area of ahyperbolic sector. It can be shown to be equal to the corresponding area against anasymptote. The quadrature was first accomplished byGregoire de Saint-Vincent in 1647 inOpus geometricum quadrature circuli et sectionum coni. As expressed by a historian,

[He made the] quadrature of a hyperbola to itsasymptotes, and showed that as thearea increased inarithmetic series theabscissas increased ingeometric series.^[5]

A. A. de Sarasa interpreted the quadrature as alogarithm and thus the geometrically definednatural logarithm (or "hyperbolic logarithm") is understood as the area undery = 1/x to the right ofx = 1. As an example of atranscendental function, the logarithm is more familiar than its motivator, the hyperbolic angle. Nevertheless, the hyperbolic angle plays a role when thetheorem of Saint-Vincent is advanced withsqueeze mapping.

Circulartrigonometry was extended to the hyperbola byAugustus De Morgan in histextbookTrigonometry and Double Algebra.^[6] In 1878W.K. Clifford used the hyperbolic angle toparametrize aunit hyperbola, describing it as "quasi-harmonic motion".

In 1894Alexander Macfarlane circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generatehyperbolic versors, in his bookPapers on Space Analysis.^[7] The following yearBulletin of the American Mathematical Society publishedMellen W. Haskell's outline of thehyperbolic functions.^[8]

WhenLudwik Silberstein penned his popular 1914 textbook on the newtheory of relativity, he used therapidity concept based on hyperbolic anglea, wheretanha =v/c, the ratio of velocityv to thespeed of light. He wrote:

It seems worth mentioning that tounit rapidity corresponds a huge velocity, amounting to 3/4 of the velocity of light; more accurately we havev = (.7616)c fora = 1.

[...] the rapiditya = 1, [...] consequently will represent the velocity .76 c which is a little above the velocity of light in water.

Silberstein also usesLobachevsky's concept ofangle of parallelism Π(a) to obtaincos Π(a) =v/c.^[9]

Imaginary circular angle

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The hyperbolic angle is often presented as if it were animaginary number, ${\textstyle \cos ix=\cosh x}$ and ${\textstyle \sin ix=i\sinh x,}$ so that thehyperbolic functions 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, ${\textstyle \cosh ix=\cos x}$ and ${\textstyle \sinh ix=i\sin x.}$

These relationships can be understood in terms of theexponential function, which for a complex argument ${\textstyle z}$ can be broken intoeven and odd parts ${\textstyle \cosh z={\tfrac {1}{2}}(e^{z}+e^{-z})}$ and ${\textstyle \sinh z={\tfrac {1}{2}}(e^{z}-e^{-z}),}$ respectively. Then

$e^{z}=\cosh z+\sinh z=\cos(iz)-i\sin(iz),$

or if the argument is separated into real and imaginary parts ${\textstyle z=x+iy,}$ the exponential can be split into the product of scaling ${\textstyle e^{x}}$ and rotation ${\textstyle e^{iy},}$

$e^{x+iy}=e^{x}e^{iy}=(\cosh x+\sinh x)(\cos y+i\sin y).$

Asinfinite series,

${\begin{alignedat}{3}e^{z}&=\,\,\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}&&=1+z+{\tfrac {1}{2}}z^{2}+{\tfrac {1}{6}}z^{3}+{\tfrac {1}{24}}z^{4}+\dots \\\cosh z&=\sum _{k{\text{ even}}}{\frac {z^{k}}{k!}}&&=1+{\tfrac {1}{2}}z^{2}+{\tfrac {1}{24}}z^{4}+\dots \\\sinh z&=\,\sum _{k{\text{ odd}}}{\frac {z^{k}}{k!}}&&=z+{\tfrac {1}{6}}z^{3}+{\tfrac {1}{120}}z^{5}+\dots \\\cos z&=\sum _{k{\text{ even}}}{\frac {(iz)^{k}}{k!}}&&=1-{\tfrac {1}{2}}z^{2}+{\tfrac {1}{24}}z^{4}-\dots \\i\sin z&=\,\sum _{k{\text{ odd}}}{\frac {(iz)^{k}}{k!}}&&=i\left(z-{\tfrac {1}{6}}z^{3}+{\tfrac {1}{120}}z^{5}-\dots \right)\\\end{alignedat}}$

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

Natural logarithm

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Theone-parameter group ofsqueeze mappings preserves areas.

The natural logarithm was first known ashyperbolic logarithm, whichGregorio a San Vincente posited asquadrature of a hyperbola in 1647. The particular hyperbolay = 1/x boundshyperbolic sectors which havearea that is the same after as before asqueeze mapping as shown in the animation.

A swapping in and out, of triangles of one-half unit area, shows the area of a hyperbolic sector is equal to the area of a region against an asymptote. The region represents theintegral of 1/x over the segment on the asymptote. Its value depends only on the ratio of the ends of the interval. Standard usage has 1 at one end. If the second endx is less than 1, then

\ln x=\int _{1}^{x}{\frac {dx}{x}}=-\int _{x}^{1}{\frac {dx}{x}}<0.

Leonhard Euler coined the phrasenatural logarithm in 1748 after he found e (Euler’s number) as the number giving a unit of area. Then theexponential function e^x has the natural logarithm for its inverse.