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.

Contents 1 Comparison with circular angle 2 History 3 Notes 4 References

[edit] Comparison with circular angle

Hyperbolic angles can be motivated by considering a geometry (now usually called Pseudoeuclidean or Minkowski geometry) where, unlike Euclidean geometry, two directions are considered to be orthogonal not if they meet at 90° as in Euclidean geometry, but if they have opposite slopes.

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

Circular angles can be characterised geometrically by the property that the if two chords P₀P₁ and P₀P₂ subtend angles L₁ and L₂ at the centre of a circle, their sum L₁ + L₂ is the angle subtended by a chord PQ, where PQ is required to be parallel to P₁P₂.

The same construction can also be applied to the hyperbola. If P₀ is taken to be the point (1,1), P₁ the point (x₁,1/x₁), and P₂ the point (x₂,1/x₂), then the parallel condition requires that Q be the point (x₁x₂,1/x₁x₂). It thus makes sense to define the hyperbolic angle from P₀ to an arbitrary point on the curve as a logarithmic function of the point's value of x. ^[1]

Just as a point on a circular segment sweeps out an area proportional to the circular angle, so too it can be shown that with the above definition a point on a hyperbolic sector sweeps out an area proportional to the hyperbolic angle. A circle centred at (0,0) passing through (1,1) accumulates an area of 2π (since its radius is ). The same area units can also be used to give a scale to hyperbolic angles. It is notable that the hyperbola through (1,1) is always as far or further from the origin than the circle, so the hyperbolic angle for any ray emerging from the origin is always as large or larger than the same ray's circular angle.

[edit] History

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 little 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] Notes

[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.
• Arthur Kennelly (1912) Application of hyperbolic functions to electrical engineering problems