A hyperbolic sector is a

region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...

of the

Cartesian plane In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

bounded by a

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

and two rays from the origin to it. For example, the two points and on the

rectangular hyperbola In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirro ...

, or the corresponding region when this hyperbola is re-scaled and its

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...

is altered by a

rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...

leaving the center at the origin, as with the

unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative rad ...

. A hyperbolic sector in standard position has and . The argument of

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...

s is the

hyperbolic angle 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 parametrizes the unit hyperbola, which has hyperbolic functio ...

, which is defined as the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a hyperbolic sector of the standard hyperbola ''xy'' = 1. This area is evaluated using

natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

Hyperbolic triangle

When in standard position, a hyperbolic sector determines a hyperbolic triangle, the

right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

with one vertex at the origin, base on the diagonal ray ''y'' = ''x'', and third vertex on the

xy=1,\,

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 :

\sqrt 2 \cosh u,\,

and the

altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...

is :

\sqrt 2 \sinh u,\,

where ''u'' is the appropriate

. The usual definitions of the hyperbolic functions can be seen via the legs of right triangles plotted with

hyperbolic coordinates In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane :\ = Q. Hyperbolic coordinates take values in the hyperbolic plane defined as: :HP = \. These coordinates in ''HP'' are useful for s ...

. When the length of theses legs is divided by the

square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...

, they can be graphed as the

with hyperbolic cosine and sine coordinates. The analogy between circular and hyperbolic functions was described by

Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He is best known for De Morgan's laws, relating logical conjunction, disjunction, and negation, and for coining the term "mathematical induction", the ...

in his ''Trigonometry and Double Algebra'' (1849).

William Burnside :''This English mathematician is sometimes confused with the Irish mathematician William S. Burnside (1839–1920).'' __NOTOC__ William Burnside (2 July 1852 – 21 August 1927) was an English mathematician. He is known mostly as an early rese ...

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".

Hyperbolic logarithm

It is known that f(''x'') = ''x''^''p'' has an algebraic

antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a continuous function is a differentiable function whose derivative is equal to the original function . This can be stated ...

except in the case ''p'' = –1 corresponding to the quadrature of the hyperbola. The other cases are given by

Cavalieri's quadrature formula In calculus, Cavalieri's quadrature formula, named for 17th-century Italian mathematician Bonaventura Cavalieri, is the integral :\int_0^a x^n\,dx = \tfrac\, a^ \qquad n \geq 0, and generalizations thereof. This is the definite integral form; ...

. Whereas quadrature of the parabola had been accomplished by

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

in the third century BC (in ''

The Quadrature of the Parabola ''Quadrature of the Parabola'' () is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing t ...

''), 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 ''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...

, the natural logarithm was known in terms of the area of a hyperbolic sector.

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

changed that when he introduced

transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction ...

s 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 In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

to the transcendental function e^x. Proposition: Given 0 < ''a'' < ''b'' and P = (''a'', 1/''a''), Q = (''b'', 1/''b''), the area of the hyperbolic sector POQ is log b/a. :proof: In the figure, POQ = POS + PQRS − QOR. Then the equality of areas POS and QOR implies area POQ = area PQRS =

\int_a^b \frac = \log b - \log a = log \frac

. In particular, for a hyperbolic sector in standard position (''a'' = 1), the area of the hyperbolic sector is log ''b''.

Hyperbolic geometry

When

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

's book on

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...

was published in 1928, it provided a foundation for the subject by reference to

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

. To establish hyperbolic measure on a line, Klein noted that the area of a hyperbolic sector provided visual illustration of the concept. Hyperbolic sectors can also be drawn to the hyperbola

y = \sqrt

. The area of such hyperbolic sectors has been used to define hyperbolic distance in a geometry textbook.Jürgen Richter-Gebert (2011) ''Perspectives on Projective Geometry'', p. 385,

References

{{reflist * Mellen W. Haskell (1895
On the introduction of the notion of hyperbolic functions

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

1(6):155–9. Area Elementary geometry Integral calculus Logarithms Euclidean plane geometry

Hyperbolic triangle

Hyperbolic logarithm

Hyperbolic geometry

See also

References