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| [[Image:Hyperbolic sector.svg|200px|right]]
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| 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 [[hyperbola]] ''xy'' = 1.
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| A hyperbolic sector in standard position has ''a'' = 1 and ''b'' > 1 .
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| ==Area==
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| The [[area]] of a hyperbolic sector in standard position is [[natural logarithm | log<sub>e</sub>]] ''b'' . | |
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| 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'')}.
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| <ref>V.G. Ashkinuse & [[Isaak Yaglom]] (1962) ''Ideas and Methods of Affine and Projective Geometry'' (in [[Russian language|Russian]]), page 151, Ministry of Education, Moscow</ref>
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| When in standard position, a hyperbolic sector corresponds to a positive [[hyperbolic angle]].
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| ==Hyperbolic triangle==
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| [[File:Cartesian hyperbolic triangle.svg|right|250px|thumb|'''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 function|hyperbolic cosine and sine functions]].]]
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| When in standard position, a hyperbolic sector determines a '''hyperbolic triangle''', the [[right triangle]] with one [[vertex (geometry)|vertex]] at the origin, base on the diagonal ray ''y'' = ''x'', and third vertex on the [[hyperbola]]
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| :<math>xy=1.\,</math>
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| The length of the base of such a triangle is
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| :<math>\sqrt 2 \cosh u,\,</math>
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| and the [[altitude (triangle)|altitude]] is
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| :<math>\sqrt 2 \sinh u,\,</math>
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| where ''u'' is the appropriate [[hyperbolic angle]].
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| The analogy between circular and hyperbolic functions was described by [[Augustus De Morgan]] in his ''Trigonometry and Double Algebra'' (1849).<ref>Augustus De Morgan (1849) [http://books.google.com/books?id=7UwEAAAAQAAJ ''Trigonometry and Double Algebra''], Chapter VI: "On the connection of common and hyperbolic trigonometry"</ref> [[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".<ref>William Burnside (1890) [[Messenger of Mathematics]] 20:145–8, see diagram page 146</ref>
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| ==Hyperbolic logarithm==
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| [[Image:hyperbola E.svg|thumb|Unit area for ''x'' = e]]
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| Students of [[integral calculus]] know that f(''x'') = ''x''<sup>p</sup> 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 3rd century BC ([[The Quadrature of the Parabola]]), the hyperbolic quadrature required the invention of a new function: [[Gregoire de Saint-Vincent]] addressed the problem of computing the area of a hyperbolic sector. 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.
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| The natural logarithm is a [[transcendental function]], an entity beyond the class of [[algebraic function]]s. Evidently transcendental functions are necessary in integral calculus.
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| ==Hyperbolic geometry==
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| {{Main|Hyperbolic geometry}}
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| 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.<ref>[[Felix Klein]] (1928) ''Vorlesungen über Nicht-Euklidische Geometrie'', p. 173, figure 113, [[Julius Springer]], Berlin</ref>
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| Hyperbolic sectors can also be drawn to the hyperbola <math>y = \sqrt{1 + x^2}</math>. The area of such hyperbolic sectors has been used to define hyperbolic distance in a geometry [[textbook]].<ref>Jürgen Richter-Gebert (2011) ''Perspectives on Projective Geometry'', p. 385, ISBN 9783642172854 {{MR|id=2791970}}</ref>
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| == See also ==
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| * [[Squeeze mapping]]
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| ==References==
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| {{reflist}}
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| [[Category:Area]]
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| [[Category:Elementary geometry]]
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| [[Category:Integral calculus]]
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| [[Category:Logarithms]]
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