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Wikipedia / Hyperbolic angle

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. Hyperbolic angle is a shuffled form of natural logarithm as they both are defined as an area against hyperbola xy = 1, and they both are preserved by squeeze mappings since those mappings preserve area.

The hyperbola xy = 1 is rectangular, analogous to the circular angle equaling the area of a circular sector.

Hyperbolic angle is used as the 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 triangle. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates.

Definition

.

First define:

Note that by properties of natural logarithm:

  • Unlike circular angle, the hyperbolic angle is unbounded is unbounded); this is related to the fact that the harmonic series is unbounded.

  • , the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is directed.

Finally, extend the definition of hyperbolic angle are positive real numbers and determine an interval on it. Then the squeeze mapping to the standard position. By the result of Gregoire de Saint-Vincent.

Comparison with circular angle

A unit circle has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola 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 sections, and hence are treated as projective ranges 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 chordss PP and PP subtend angles L and L at the centre of a circle, their sum is the angle subtended by a chord PQ, where PQ is required to be parallel to PP.

The same construction can also be applied to the hyperbola. If P is taken to be the point , P the point , and P the point , then the parallel condition requires that Q be the point . It thus makes sense to define the hyperbolic angle from P to an arbitrary point on the curve as a logarithmic function of the point's value of x.

Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a 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.

Both circular and hyperbolic angle provide instances of an invariant measure. Arcs with an angular magnitude on a circle generate a measure on certain measurable sets on the circle whose magnitude does not vary as the circle turns or 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

  1. (x, y) ↦ (rx, y / r), with r > 0 .

Relation To The Minkowski Line Element

There is also a relation to a hyperbolic angle and the metric defined on Minkowski space. Just as two dimensional Euclidean geometry defines its line element as

the line element on Minkowski space is

Consider a curve embedded in two dimensional Euclidean space,

). The arclength of this curve in Euclidean space is computed as:

. Now doing the same procedure, except replacing the Euclidean element with the Minkowski line element,

. 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 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,

  1. [He made the] quadrature of a hyperbola to its asymptotes, and showed that as the area increased in arithmetic series the abscissas increased in geometric series.

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 to the right of . 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 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. In 1878 W.K. Clifford used the hyperbolic angle to parametrize a unit hyperbola, describing it as "quasi-harmonic motion".

In 1894 Alexander Macfarlane circulated his essay "The Imaginary of Algebra", which used hyperbolic angles to generate hyperbolic versors, in his book Papers on Space Analysis. The following year Bulletin of the American Mathematical Society published Mellen W. Haskell's outline of the hyperbolic functions.

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 , the ratio of velocity v to the speed of light. He wrote:

  1. 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 for .

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

Silberstein also uses Lobachevsky's concept of angle of parallelism Π(a) to obtain .

Imaginary circular angle

The hyperbolic angle is often presented as if it were an imaginary number so that the hyperbolic functions

These relationships can be understood in terms of the exponential function can be broken into even and odd parts respectively. Then

As infinite series,

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.

Natural logarithm

The natural logarithm was first known as hyperbolic logarithm, which Gregorio a San Vincente posited as quadrature of a hyperbola in 1647. The particular hyperbola y = 1/x bounds hyperbolic sectors which have area that is the same after as before a squeeze 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 the integral 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 end x is less than 1, then

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

See also

Notes

References