Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that defines a hyperbolic sector. The relationship of a hyperbolic angle to a hyperbola parallels the relationship of an "ordinary" angle to a circle.
The magnitude of the hyperbolic angle is the area of the corresponding sector of the hyperbola "xy" = 1. This hyperbola is rectangular with a semi-major axis of formula_1, analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius formula_1.
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 parameter thus becomes one of the most useful in the calculus of real variables.
Definition.
Consider the rectangular hyperbola formula_3, and (by convention) pay particular attention to the "branch" formula_4.
First define:
Note that, because of the role played by the natural logarithm:
Finally, extend the definition of "hyperbolic angle" to that subtended by any interval on the hyperbola. Suppose formula_12 are positive real numbers such that formula_13 and formula_14, so that formula_15 and formula_16 are points on the hyperbola formula_17 and determine an interval on it. Then the squeeze mapping formula_18 maps the angle formula_19 to the "standard position" angle formula_20. 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 formula_21.
Comparison with circular angle.
A unit circle formula_22 has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola formula_23 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 characterised geometrically by the property that if two chords "P"0"P"1 and "P"0"P"2 subtend angles "L"1 and "L"2 at the centre of a circle, their sum is the angle subtended by a chord "PQ", where "PQ" is required to be parallel to "P"1"P"2.
The same construction can also be applied to the hyperbola. If "P"0 is taken to be the point , "P"1 the point , and "P"2 the point , then the parallel condition requires that "Q" be the point . It thus makes sense to define the hyperbolic angle from "P"0 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
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 his momentous "Opus geometricum quadrature circuli et sectionum coni". As expressed by a historian,
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:
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. Thus, if "x" is a real number and , then
so that the hyperbolic functions cosh and sinh can be presented through the circular functions. But these identities do not arise from a circle or rotation, rather they can be understood in terms of infinite series. In particular, the one expressing the exponential function (formula_25 ) consists of even and odd terms, the former comprise the cosh function (formula_26), the latter the sinh function (formula_27). 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. The above identities use the number "i" to remove the alternating factor (−1)"n" from terms of the series to restore the full halves of the exponential series. Nevertheless, in the theory of holomorphic functions, the hyperbolic sine and cosine functions are incorporated into the complex sine and cosine functions.