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Concept
Hyperbolic sector
Summary
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola. A hyperbolic sector in standard position has a = 1 and b > 1.
Hyperbolic sectors are the basis for the hyperbolic functions.
The area of a hyperbolic sector in standard position is natural logarithm of b .
Proof: Integrate under 1/x from 1 to b, add triangle {(0, 0), (1, 0), (1, 1)}, and subtract triangle {(0, 0), (b, 0), (b, 1/b)} (both triangles of which have the same area).
When in standard position, a hyperbolic sector corresponds to a positive hyperbolic angle at the origin, with the measure of the latter being defined as the area of the former.
When in standard position, a hyperbolic sector determines a hyperbolic triangle, the right triangle with one vertex at the origin, base on the diagonal ray y = x, and third vertex on the hyperbola
with the hypotenuse being the segment from the origin to the point (x, y) on the hyperbola. The length of the base of this triangle is
and the altitude is
where u is the appropriate hyperbolic angle.
The analogy between circular and hyperbolic functions was described by Augustus De Morgan in his Trigonometry and Double Algebra (1849). William Burnside used such triangles, projecting from a point on the hyperbola xy = 1 onto the main diagonal, in his article "Note on the addition theorem for hyperbolic functions".
Natural logarithm
It is known that f(x) = xp has an algebraic antiderivative except in the case p = –1 corresponding to the quadrature of the hyperbola. The other cases are given by Cavalieri's quadrature formula. Whereas quadrature of the parabola had been accomplished by Archimedes in the third century BC (in The Quadrature of the Parabola), the hyperbolic quadrature required the invention in 1647 of a new function: Gregoire de Saint-Vincent addressed the problem of computing the areas bounded by a hyperbola.
Official source
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