Hyperbolic sector

A hyperbolic sector is a region of the Cartesian plane {(x,y)} bounded by rays from the origin to two points (a, 1/a) and (b, 1/b) and by the rectangular hyperbola xy = 1 (or the corresponding region when this hyperbola is rescaled 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.

Area

The area of a hyperbolic sector in standard position is ln 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)}. ^[1]

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.

Hyperbolic triangle

Hyperbolic triangle (yellow) and hyperbolic sector (red) corresponding to hyperbolic angle u, to the rectangular hyperbola (equation y = 1/x). The legs of the triangle are √2 times the hyperbolic cosine and sine functions.

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).^[2] 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".^[3]

Hyperbolic logarithm

Unit area when b = e as exploited by Euler.

Students of integral calculus know that f(x) = x^p 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. His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.

Before 1748 and the publication of Introduction to the Analysis of the Infinite, the natural logarithm was known in terms of the area of a hyperbolic sector. Leonhard Euler changed that when he introduced transcendental functions such as 10^x. Euler identified e as the value of b producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position). Then the natural logarithm could be recognized as the inverse function to the transcendental e^x.

Hyperbolic geometry

When Felix Klein wrote his book on non-Euclidean geometry in 1928, he provided a foundation for the subject by reference to projective geometry. To establish hyperbolic measure on a line, he noted that the area of a hyperbolic sector provided visual illustration of the concept.^[4]

Hyperbolic sectors can also be drawn to the hyperbola . The area of such hyperbolic sectors has been used to define hyperbolic distance in a geometry textbook.^[5]

See also

• Squeeze mapping

References

• Mellen W. Haskell (1895) On the introduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1(6):155–9.