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Hyperbolic sector
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{{Short description|Region of the Cartesian plane bounded by a hyperbola and two radii}} [[File:Hyperbolic_sector.svg|right|frameless]] A '''hyperbolic sector''' is a [[region (mathematics)|region]] of the [[Cartesian plane]] bounded by a [[hyperbola]] and two [[ray (geometry)|rays]] from the origin to it. For example, the two points {{math|(''a'', 1/''a'')}} and {{math|(''b'', 1/''b'')}} on the [[Hyperbola#Rectangular hyperbola|rectangular hyperbola]] {{math|1=''xy'' = 1}}, or the corresponding region when this hyperbola is re-scaled and its [[orientation (geometry)|orientation]] is altered by a [[Rotation (geometry)|rotation]] leaving the center at the origin, as with the [[Unit hyperbola#Parametrization|unit hyperbola]]. A hyperbolic sector in standard position has {{math|1=''a'' = 1}} and {{math|1=''b'' > 1}}. Hyperbolic sectors are the basis for the [[hyperbolic function]]s. ==Area== [[File:Hyperbolic sector squeeze mapping.svg|thumb|Hyperbolic sector area is preserved by [[squeeze mapping]], shown squeezing rectangles and rotating a hyperbolic sector]] 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). <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> 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== [[File:Cartesian hyperbolic triangle.svg|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 {{radic|2}} times the [[Hyperbolic function|hyperbolic cosine and sine functions]].]] 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]] :<math>xy=1,\,</math> 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 :<math>\sqrt 2 \cosh u,\,</math> and the [[altitude (triangle)|altitude]] is :<math>\sqrt 2 \sinh u,\,</math> where ''u'' is the appropriate [[hyperbolic angle]]. The usual definitions of the hyperbolic functions can be seen via the legs of right triangles plotted with [[hyperbolic coordinates#Trigonometry|hyperbolic coordinates]]. When the length of theses legs is divided by the [[square root of 2]], they can be graphed as the [[unit hyperbola]] with hyperbolic cosine and sine coordinates. 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) [https://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> ==Hyperbolic logarithm== [[Image:hyperbola E.svg|thumb|Unit area when ''b'' = ''e'' as exploited by Euler.]] {{Main article|Natural logarithm}} It is known that f(''x'') = ''x''<sup>''p''</sup> has an algebraic [[antiderivative]] except in the case ''p'' = –1 corresponding to the [[quadrature (mathematics)|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.<ref>Martin Flashman [http://users.humboldt.edu/flashman/Presentations/HSU%20Colloquia/colloq2_4_99.html The History of Logarithms] from [[Humboldt State University]] </ref> 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 function]]s such as 10<sup>x</sup>. Euler identified [[e (mathematical constant)|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 function e<sup>x</sup>. To accommodate the case of negative logarithms and the corresponding negative hyperbolic angles, different hyperbolic sectors are constructed according to whether ''x'' is greater or less than one. A variable right triangle with area 1/2 is <math>V = \{(x, 1/x), \ (x,0), \ (0,0)\} .</math> The isosceles case is <math>T = \{(1,1),\ (1,0),\ (0,0)\}.</math> The natural logarithm is known as the area under ''y'' = 1/''x'' between one and ''x''. A positive hyperbolic angle is given by the area of <math>\int_1^x \frac{dt}{t} + T - V.</math> A negative hyperbolic angle is given by the ''negative'' of the area <math>\int_x^1 \frac{dt}{t} + V - T.</math> This convention is in accord with a negative natural logarithm for ''x'' in (0,1). ==Hyperbolic geometry== {{Main article|Hyperbolic geometry}} When [[Felix Klein]]'s book on [[non-Euclidean geometry]] was published in 1928, it provided a foundation for the subject by reference to [[projective geometry]]. To establish hyperbolic measure on a line, Klein 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> 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> == See also == * [[Squeeze mapping]] ==References== {{reflist}} * [[Mellen W. Haskell]] (1895) [https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf On the introduction of the notion of hyperbolic functions] [[Bulletin of the American Mathematical Society]] 1(6):155–9. [[Category:Area]] [[Category:Elementary geometry]] [[Category:Integral calculus]] [[Category:Logarithms]] [[Category:Euclidean plane geometry]]
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