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A hyperbolic sector is a
region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
of the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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 ra ...
. A hyperbolic sector in standard position has and . Hyperbolic sectors are the basis for the
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 u ...
s.


Area

The
area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 op ...
of a hyperbolic sector in standard position is
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of ''b'' . Proof: Integrate under 1/''x'' from 1 to ''b'', add triangle , and subtract triangle . When in standard position, a hyperbolic sector corresponds to a positive
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 parametrises the unit hyperbola, which has hyperbolic function ...
at the origin, with the measure of the latter being defined as the area of the former.


Hyperbolic triangle

When in standard position, a hyperbolic sector determines a hyperbolic triangle, the
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
with one
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
at the origin, base on the diagonal ray ''y'' = ''x'', and third vertex on the
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) 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, ca ...
: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 or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
is :\sqrt 2 \sinh u,\, where ''u'' is the appropriate
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 parametrises the unit hyperbola, which has hyperbolic function ...
. The analogy between circular and hyperbolic functions was described by Augustus De Morgan 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 res ...
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 function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
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 Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...
in the third century BC (in ''
The Quadrature of the Parabola ''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions rega ...
''), 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 Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
changed that when he introduced
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed ...
s such as 10x. 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 X ...
to the transcendental function ex.


Hyperbolic geometry

When
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
wrote his 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 g ...
in 1928, he 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, ...
. To establish hyperbolic measure on a line, he 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,


See also

* Squeeze mapping


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