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Concept
Hyperbolic angle
Summary
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. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
The hyperbola xy = 1 is rectangular with a semi-major axis of , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius .
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.
Consider the rectangular hyperbola , and (by convention) pay particular attention to the branch .
First define:
The hyperbolic angle in standard position is the angle at between the ray to and the ray to , where .
The magnitude of this angle is the area of the corresponding hyperbolic sector, which turns out to be .
Note that, because of the role played by the natural logarithm:
Unlike the circular angle, the hyperbolic angle is unbounded (because is unbounded); this is related to the fact that the harmonic series is unbounded.
The formula for the magnitude of the angle suggests that, for , the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is directed.
Finally, extend the definition of hyperbolic angle to that subtended by any interval on the hyperbola. Suppose are positive real numbers such that and , so that and are points on the hyperbola and determine an interval on it. Then the squeeze mapping maps the angle to the standard position angle . 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.
Official source
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