- Hyperbolic coordinates
In
mathematics , hyperbolic coordinates are a useful method of locating points in Quadrant I of theCartesian plane :x, y) : x > 0, y > 0 } = Q !.
Hyperbolic coordinates take values in
:HP = {(u, v) : u in mathbb{R}, v > 0 }.
For x,y) in Q take
:u = -frac{1}{2} log left( frac{y}{x} ight)
and
:v = sqrt{xy}.
Sometimes the parameter u is called
hyperbolic angle and v thegeometric mean .The inverse mapping is
:x = v e^u ,quad y = v e^{-u}.
This is a
continuous mapping , but not ananalytic function .Quadrant model of hyperbolic geometry
The correspondence
:Q leftrightarrow HP
affords the
hyperbolic geometry structure to "Q" that is erected on "HP" byhyperbolic motion s. The "hyperbolic lines" in "Q" are rays from the origin orpetal -shapedcurve s leaving and re-entering the origin. The left-right shift in "HP" corresponds to asqueeze mapping applied to "Q".Applications in physical science
Physical unit relations like:
* E = IR :Ohm's law
* P = EI :Electrical power
* PV = kT :Ideal gas law all suggest looking carefully at the quadrant. For example, inthermodynamics theisothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, an isobaric process may trace a hyperbola in the quadrant of absolute temperature and gas density.tatistical applications
*Comparative study of
population density in the quadrant begins with selecting a reference nation, region, or urban area whose population and area are taken as the point (1,1).
*Analysis of the elected representation of regions in arepresentative 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 currencyexchange rate fluctuation:The unit currency sets x = 1. The price currency corresponds to y. For:0 < y < 1
we find u > 0, a positive hyperbolic angle. For a "fluctuation" take a new price
:0 < z < y.
Then the change in "u" is:
:Delta u = frac{1}{2} log left( frac{y}{z} ight).
Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity Delta u 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 abasket of consumer goods .
* Quantification of change in marketshare induopoly .
* Corporatestock split s versus stock buy-back.
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