Hyperbolic angle

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A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1.

The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is log_e x.

Note that unlike circular angle, hyperbolic angle is unbounded, as is the function log_e x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle is considered to be negative when 0 < x < 1.

The hyperbolic functions sinh, cosh, and tanh use the hyperbolic angle as their independent variable because their values may be premised on analogies to circular trigonometric functions when the hyperbolic angle defines a hyperbolic triangle. Thus this parameter becomes one of the most useful in the calculus of a real variable.

The quadrature of the hyperbola is the evaluation of the area swept out by a radial segment from the origin as the terminus moves along the hyperbola, just the topic of hyperbolic angle. The quadrature of the hyperbola was first accomplished by Gregoire de Saint-Vincent in 1647 in his momentous Opus geometricum quadrature circuli et sectionum coni. As David Eugene Smith wrote in 1925:

[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.

History of Mathematics, pp. 424,5 v. 1

The upshot was the logarithm function, as now understood as the area under y = 1/x to the right of 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 theorem of Saint-Vincent is advanced with squeeze mapping.

When Ludwik Silberstein penned his popular textbook on the new theory of relativity, he used the rapidity concept based on hyperbolic angle a where 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 v = (.7616) c for a = 1.

... the rapidity a = 1, ... consequently will represent the velocity .76 c which is a litle above the velocity of light in water.

Silberstein also uses Lobachevsky's concept of angle of parallelism Π(a) to obtain cos Π(a) = v/c.

[edit] References

• John Stillwell (1998) Numbers and Geometry exercise 9.5.3, p. 298, Springer-Verlag ISBN 0-387-98289-2.
• Ludwik Silberstein (1914) Theory of Relativity, Cambridge University Press, pp.180-1.