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en>ChrisGualtieri: General Fixes using AWB

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General Fixes using AWB

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	There is ~~nothing~~ to ~~write about me~~ at ~~all~~.<br>~~Yes! Im~~ a ~~member~~ of this ~~site~~.<br>~~I really hope I~~'~~m useful~~ in ~~some way~~ .<br><br>~~Take~~ a ~~look~~ at ~~my web~~-~~site~~ [http://~~minecraftminigamesonline~~.~~com~~/~~profile~~/~~356076~~/~~njuep type my essay online~~]		[[File:Drini-conjugatehyperbolas.svg\|thumb\|right\|The Unit Hyperbola is blue, its conjugate is green, and the asymptotes are red.]]
			In [[geometry]], the '''unit hyperbola''' is the set of points (''x,y'') in the [[Cartesian plane]] that satisfies <math> x^2 - y^2 = 1 .</math> In the study of [[indefinite orthogonal group]]s, the unit hyperbola forms the basis for an ''alternative radial length''
			: <math>r = \sqrt {x^2 - y^2} .</math>
			Whereas the [[unit circle]] surrounds its center, the unit hyperbola requires the ''conjugate hyperbola'' <math>y^2 - x^2 = 1 </math> to complement it in the plane. This pair of hyperbolas share the [[asymptote]]s ''y'' = ''x'' and ''y'' = −''x''. When the conjugate of the unit hyperbola is in use, the alternative radial length is <math>r = \sqrt {y^2 - x^2} . </math>

			The unit hyperbola is a special case of the [[rectangular hyperbola]], with a particular [[Orientation (geometry)\|orientation]], [[translation (geometry)\|location]], and [[Scaling (geometry)\|scale]]. As such, its [[Eccentricity (mathematics)\|eccentricity]] equals <math>\sqrt{2}.</math>

			The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of [[spacetime]] as a [[pseudo-Euclidean space]]. There the asymptotes of the unit hyperbola form a [[light cone]]. Further, the attention to areas of [[hyperbolic sector]]s by [[Gregoire de Saint-Vincent]] led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical [[complex number]]s, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola.

			==Asymptotes==
			{{main\|asymptote}}
			Generally asymptotic lines to a curve are said to converge toward the curve. In [[algebraic geometry]] and the theory of [[algebraic curves]] there is a different approach to asymptotes. The curve is first interpreted in the [[projective plane]] using [[homogeneous coordinates]]. Then the asymptotes are lines that are tangent to the projective curve at a [[point at infinity]], thus circumventing any need for a distance concept and convergence. In a common framework (''x, y, z'') are homogeneous coordinates with the [[line at infinity]] determined by the equation ''z'' = 0. For instance, C. G. Gibson wrote:<ref>C.G. Gibson (1998) ''Elementary Geometry of Algebraic Curves'', p 159, [[Cambridge University Press]] ISBN 0-521-64140-3</ref>
			:For the standard rectangular hyperbola <math>\scriptstyle f = x^2 - y^2 -1</math> in R<sup>2</sup> the corresponding projective curve is <math>\scriptstyle F = x^2 - y^2 - z^2,</math> which meets ''z'' = 0 at the points ''P'' = (1 : 1 : 0) and ''Q'' = (1 : −1 : 0). Both ''P, Q'' are [[zero (complex analysis)#Multiplicity of a zero\|simple]] on ''F'', with tangents ''x'' + ''y'' = 0, ''x'' − ''y'' = 0; thus we recover the familiar 'asymptotes' of elementary geometry.

			==Minkowski diagram==
			{{main\|Minkowski diagram}}
			The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are
			* units of 30 centimetres length and [[nanosecond]]s, or
			* [[astronomical unit]]s and intervals of 8 minutes and 20 seconds, or
			* [[light year]]s and [[year]]s.
			Each of these scales of coordinates results in [[photon]] connections of events along diagonal lines of [[slope]] plus or minus one.
			Five elements constitute the diagram [[Hermann Minkowski]] used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the [[conjugate diameters\|conjugate diameter]].
			The plane with the axes refers to a resting [[frame of reference]]. The diameter of the unit hyperbola represents a frame of reference in motion with [[rapidity]] a where <math>\scriptstyle \tanh \ a = y/x</math> and (''x,y'') is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents the ''spatial hyperplane of simultaneity'' corresponding to rapidity ''a''.
			In this context the unit hyperbola is a ''calibration hyperbola''<ref>[[Anthony French]] (1968) ''Special Relativity'', page 83, [[W. W. Norton & Company]]</ref><ref>W.G.V. Rosser (1964) ''Introduction to the Theory of Relativity'', figure 6.4, page 256, London: [[Butterworths]]</ref>

			Commonly in relativity study the hyperbola with vertical axis is taken as primary:
			:The arrow of time goes from the bottom to top of the figure — a convention adopted by [[Richard Feynman]] in his famous dagrams. Space is represented by planes perpendicular to the time axis. The here and now is a singularity in the middle.<ref>A.P. French (1989) "Learning from the past; Looking to the future", acceptance speech for 1989 [[Oersted Medal]], [[American Journal of Physics]] 57(7):587–92</ref>
			The vertical time axis convention stems from Minkowski in 1908, and
			is also illustrated on page 48 of Eddington's ''The Nature of the Physical World'' (1928).

			==Parametrization==
			{{main\|hyperbolic angle}}
			As a particular [[conic section\|conic]], the hyperbola can be parametrized by the process of addition of points on a conic. The following description
			was given by Russian analysts:
			:Fix a point ''E'' on the conic. Consider the points at which the straight line drawn through ''E'' parallel to ''AB'' intersects the conic a second time to be the ''sum of the points A and B''.
			:For the hyperbola <math>x^2 - y^2 = 1</math> with the fixed point ''E'' = (1,0) the sum of the points <math>(x_1,\ y_1)</math> and <math>(x_2,\ y_2)</math> is the point <math>(x_1 x_2 + y_1 y_2,\ y_ 1 x_2 + y_2 x_1 )</math> under the parametrization <math>x = \cosh \ t</math> and <math>y = \sinh \ t</math> this addition corresponds to the addition of the parameter ''t''.<ref>Viktor Prasolov & Yuri Solovyev (1997) ''Elliptic Functions and Elliptic Integrals'', page one, Translations of Mathematical Monographs volume 170, [[American Mathematical Society]]</ref>

			[[Image:Hyperbolic functions-2.svg\|thumb\|296px\|right\|The branches of the unit hyperbola evolve as the points (cosh α, sinh α) and (−cosh α, −sinh α) depending on the hyperbolic angle parameter α]]

			This parameter is '''hyperbolic angle''', which is the [[argument of a function\|argument]] of the [[hyperbolic function]]s.

			One finds an early expression of the parametrized unit hyperbola in [[Elements of Dynamic]] (1878) by [[William Kingdon Clifford\|W. K. Clifford]]. He describes quasi-harmonic motion in a hyperbola as follows:
			:The motion <math>\rho = \alpha \cosh(nt + \epsilon) + \beta \sinh(nt + \epsilon)</math> has some curious analogies to elliptic harmonic motion. ... The acceleration <math>\ddot{\rho} = n^2 \rho \ ;</math>  thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed ''away'' from the centre.<ref>[[William Kingdon Clifford]] (1878) [http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=04370002 Elements of Dynamic], pages 89 & 90, London: MacMillan & Co; on-line presentation by [[Cornell University]] ''Historical Mathematical Monographs''</ref>

			A direct way to parameterizing the unit hyperbola starts with the hyperbola ''xy'' = 1 parameterized with the [[exponential function]]: <math>( e^t, \ e^{-t}).</math>

			This hyperbola is transformed into the unit hyperbola by a [[linear mapping]] having the matrix <math>A = \tfrac {1}{2}\begin{pmatrix}1 & 1 \\ 1 & -1 \end{pmatrix}\ :</math>
			:<math>(e^t, \ e^{-t}) \ A = (\frac{e^t + e^{-t}}{2},\ \frac{e^t - e^{-t}}{2}) = (\cosh t,\ \sinh t).</math>

			==Complex plane algebra==
			{{main\|split-complex number}}
			Whereas the unit circle is associated with [[complex number]]s, the unit hyperbola is key to the ''split-complex number plane'' consisting of ''z'' = ''x'' + ''y j'' where ''j'' <sup>2</sup> = +1.
			Then ''jz = y + x j'' so that the action of ''j'' on the plane is to swap the coordinates. In particular, this action swaps the unit hyperbola with its conjugate, and also swaps pairs of [[conjugate diameters]] of the hyperbolas.

			In terms of the hyperbolic angle parameter ''a'', the unit hyperbola consists of points
			:<math>\pm(\cosh a + j \sinh a) </math> where ''j'' = (0,1).
			The right branch of the unit hyperbola corresponds to the positive coefficient. In fact, this branch is the image of the [[exponential map]] acting on the j-axis. Since
			:<math> \exp(aj) exp(bj) = \exp((a+b)j),</math>
			the branch is a [[group (mathematics)\|group]] under multiplication. Unlike the [[circle group]], this unit hyperbola group is ''not'' [[compact space\|compact]].
			Similar to the ordinary complex plane, a point, not on the diagonals, has a [[polar decomposition#Alternative planar decompositions\|polar decomposition]] using the parametrization of the unit hyperbola and the alternative radial length.

			==References==
			{{reflist}}
			* F. Reese Harvey (1990) ''Spinors and calibrations'', Figure 4.33, page 70, [[Academic Press]], ISBN 0-12-329650-1 .

			[[Category:Geometry]]
			[[Category:One]]
			[[Category:Analytic geometry]]

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en>Monkbot: Fix CS1 deprecated date parameter errors

en>ChrisGualtieri: General Fixes using AWB

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