Editing Hyperbolic coordinates

{{short description|Geometric mean and hyperbolic angle as coordinates in quadrant I}}
[[Image:Hyperbolic coordinates.svg|thumb|200px|right|Hyperbolic coordinates plotted on the Euclidean plane: all points on the same blue ray share the same coordinate value ''u'', and all points on the same red hyperbola share the same coordinate value ''v''.]]

In [[mathematics]], '''hyperbolic coordinates''' are a method of locating points in quadrant I of the [[Cartesian plane]]

:<math>\{(x, y) \ :\  x > 0,\ y > 0\ \} = Q</math>.

Hyperbolic coordinates take values in the [[hyperbolic plane]] defined as:

:<math>HP = \{(u, v) : u \in \mathbb{R}, v > 0 \}</math>.

These coordinates in ''HP'' are useful for studying [[logarithmic scale|logarithmic]] comparisons of [[direct proportion]] in ''Q'' and measuring deviations from direct proportion.

For <math>(x,y)</math> in <math>Q</math> take

:<math>u = \ln \sqrt{\frac{x}{y}} </math>

and

:<math>v = \sqrt{xy}</math>.

The parameter ''u'' is the [[hyperbolic angle]] to (''x, y'') and ''v'' is the [[geometric mean]] of ''x'' and ''y''.

The inverse mapping is

:<math>x = v e^u ,\quad y = v e^{-u}</math>.

The function <math>Q \rarr HP</math> is a [[continuous mapping]], but not an [[analytic function]].

==Alternative quadrant metric==
Since ''HP'' carries the [[metric space]] structure of the [[Poincaré half-plane model]] of [[hyperbolic geometry]], the bijective correspondence 
<math>Q \leftrightarrow HP</math> brings this structure to ''Q''. It can be grasped using the notion of [[hyperbolic motion]]s. Since [[geodesic]]s in ''HP'' are semicircles with centers on the boundary, the geodesics in ''Q'' are obtained from the correspondence and turn out to be [[Line (mathematics)#Ray|rays]] from the origin or [[petal]]-shaped [[curve]]s leaving and re-entering the origin. And the hyperbolic motion of ''HP'' given by a left-right shift corresponds to a [[squeeze mapping]] applied to ''Q''.

Since [[hyperbola]]s in ''Q'' correspond to lines parallel to the boundary of ''HP'', they are [[horocycle]]s in the metric geometry of ''Q''.

If one only considers the [[Euclidean topology]] of the plane and the topology inherited by ''Q'', then the lines bounding ''Q'' seem close to ''Q''. Insight from the metric space ''HP'' shows that the [[open set]] ''Q'' has only the [[origin (mathematics)|origin]] as boundary when viewed through the correspondence. Indeed, consider rays from the origin in ''Q'', and their images, vertical rays from the boundary ''R'' of ''HP''. Any point in ''HP'' is an infinite distance from the point ''p'' at the foot of the perpendicular to ''R'', but a sequence of points on this perpendicular may tend in the direction of ''p''. The corresponding sequence in ''Q'' tends along a ray toward the origin. The old Euclidean boundary of ''Q'' is no longer relevant.

==Applications in physical science==
Fundamental physical variables are sometimes related by equations of the form ''k'' = ''x y''. For instance, ''V'' = ''I R'' ([[Ohm's law]]), ''P'' = ''V I'' ([[electrical power]]), ''P V'' = ''k T'' ([[ideal gas law]]), and ''f'' λ = ''v'' (relation of [[wavelength]], [[frequency]], and velocity in the wave medium). When the ''k'' is constant, the other variables lie on a hyperbola, which is a [[horocycle]] in the appropriate ''Q'' quadrant.

For example, in [[thermodynamics]] the [[isothermal process]] explicitly follows the hyperbolic path and [[work (thermodynamics)|work]] can be interpreted as a hyperbolic angle change. Similarly, a given mass ''M'' of gas with changing volume will have variable density &delta; = ''M / V'', and the ideal gas law may be written ''P = k T'' &delta; so that an [[isobaric process]] traces  a hyperbola in the quadrant of absolute temperature and gas density.

For hyperbolic coordinates in the [[theory of relativity]] see the [[#History|History]] section.

==Statistical applications==
*Comparative study of [[population density]] in the quadrant begins with selecting a reference nation, region, or [[urban density|urban]] area whose population and area are taken as  the point (1,1).
*Analysis of the [[legislator|elected representation]] of regions in a [[representative democracy]] begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant.

==Economic applications==
There are many natural applications of hyperbolic coordinates in [[economics]]:
* Analysis of currency [[exchange rate]] fluctuation:{{pb}}The unit currency sets <math>x = 1</math>. The price currency corresponds to <math>y</math>. For <math display="block">0 < y < 1</math> we find <math>u > 0</math>, a positive hyperbolic angle. For a ''fluctuation'' take a new price <math display="block">0 < z < y.</math> Then the change in ''u'' is: <math display="block">\Delta u = \ln \sqrt{\frac{y}{z}}. </math> Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent [[measure (mathematics)|measure]]. The quantity <math>\Delta u</math> is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.
* Analysis of inflation or deflation of prices of a [[basket of consumer goods]].
* Quantification of change in marketshare in [[duopoly]].
* Corporate [[stock split]]s versus stock buy-back.

==Trigonometry==
[[File:Cartesian_hyperbolic_triangle.svg|thumb|right|250px|Right triangles with legs proportional to sinh and cosh]]
The [[hyperbolic function]]s sinh, cosh, and tanh can be illustrated with hyperbolic coordinates. Let
:<math>A = (e^t, e^{-t}), \ B=(e^{-t}, e^t), \ C = (e^t + e^{-t}) .</math>
Then BCAO  forms a [[rhombus]] with diagonals intersecting at <math>M = (\frac{e^t + e^{-t} }{2},\ \frac{e^t + e^{-t} }{2} ) </math>. The hyperbolic cosine is defined as <math>\cosh t = \frac{e^t + e^{-t} }{2},</math> so ''M'' = ( cosh ''t'', cosh ''t'').

The semi-diagonal MA is [[equipollence (geometry)|equipollent]] to <math>(\frac{-e^{-t} + e^t }{2}, \ \frac{e^t - e^{-t} }{2}) = (- \sinh t,\ \sinh t) </math>. Evidently the diagonals divide the rhombus into four congruent right triangles. 

The angle MOA is the [[hyperbolic angle]] parameter ''t'' of cosh and sinh, and <math>\tanh t = \frac{\sinh t}{\cosh t}</math> and has a value in (–1, 1).

==History==
The [[geometric mean]] is an ancient concept, but [[hyperbolic angle]] was developed in this configuration by [[Gregoire de Saint-Vincent]]. He was attempting to perform [[quadrature (mathematics)|quadrature]] with respect to the rectangular hyperbola ''y'' = 1/''x''. That challenge was a standing [[open problem]] since [[Archimedes]] performed the [[quadrature of the parabola]]. The curve passes through (1,1) where it is opposite the [[origin (mathematics)|origin]] in a [[unit square]]. The other points on the curve can be viewed as [[rectangle]]s having the same [[area]] as this square. Such a rectangle may be obtained by applying a [[squeeze mapping]] to the square. Another way to view these mappings is via [[hyperbolic sector]]s. Starting from (1,1) the hyperbolic sector of unit area ends at (e, 1/e), where [[e (mathematical constant)|e]] is 2.71828…, according to the development of [[Leonhard Euler]] in ''[[Introduction to the Analysis of the Infinite]]'' (1748).

Taking (e, 1/e) as the vertex of rectangle of unit area, and applying again the squeeze that made it from the unit square, yields <math>(e^2, \ e^{-2}).</math> Generally n squeezes yields <math>(e^n, \ e^{-n}).</math> [[A. A. de Sarasa]] noted a similar observation of G. de Saint Vincent, that as the abscissas increased in a [[geometric series]], the sum of the areas against the hyperbola increased in [[arithmetic series]], and this property corresponded to the '''logarithm''' already in use to reduce multiplications to additions. Euler’s work made the [[natural logarithm]] a standard mathematical tool, and elevated mathematics to the realm of [[transcendental function]]s. The hyperbolic coordinates are formed on the original picture of G. de Saint-Vincent, which provided the quadrature of the hyperbola, and transcended the limits of [[algebraic function]]s.

In 1875 [[Johann von Thünen]] published a theory of natural wages<ref>{{cite book|author=Henry Ludwell Moore|author-link=Henry Ludwell Moore|title=Von Thünen's Theory of Natural Wages|url=https://archive.org/details/vonthnenstheor00moor|year=1895|publisher=G. H. Ellis}}</ref> which used geometric mean of a subsistence wage and market value of the labor using the employer's capital.

In [[special relativity]] the focus is on the 3-dimensional [[hypersurface]] in the future of spacetime where various velocities arrive after a given [[proper time]]. Scott Walter<ref>Walter (1999) page 99</ref> explains that in November 1907 [[Hermann Minkowski]] alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.<ref>Walter (1999) page 100</ref>
In tribute to [[Wolfgang Rindler]], the author of a standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called [[Rindler coordinates]].

==References==
<references/>
*David Betounes (2001) ''Differential Equations: Theory and Applications'', page 254, Springer-TELOS, {{ISBN|0-387-95140-7}} .
*Scott Walter (1999). [http://scottwalter.free.fr/papers/1999-symbuniv-walter.html "The non-Euclidean style of Minkowskian relativity"] {{Webarchive|url=https://web.archive.org/web/20131016142709/http://www.univ-nancy2.fr/DepPhilo/walter/papers/nes.pdf |date=2013-10-16 }}. Chapter 4 in: Jeremy J. Gray (ed.), ''The Symbolic Universe: Geometry and Physics 1890-1930'', pp.&nbsp;91–127. [[Oxford University Press]]. {{ISBN|0-19-850088-2}}.

[[Category:Coordinate systems]]
[[Category:Hyperbolic geometry]]