Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola.

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 ln x.

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

Suppose ab = 1 and cd = 1 with c > a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy = 1. Then the squeeze mapping with diagonal elements b and a maps this interval to the standard position hyperbolic angle that runs from (1, 1) to (bc, ad). By the result of Gregoire de Saint-Vincent, the hyperbolic sector determined by (a, b) and (c, d) has the same area as this standard position angle, and the magnitude of the hyperbolic angle is taken to be this area.

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.

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