Hyperbolic coordinates

In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian 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 $v$ the geometric mean.

The inverse mapping is

:$x\; =\; v\; e^u\; ,quad\; y\; =\; v\; e^\{-u\}$.

This is a continuous mapping, but not an analytic function.

Quadrant model of hyperbolic geometry

The correspondence

:$Q\; leftrightarrow\; HP$

affords the hyperbolic geometry structure to "Q" that is erected on "HP" by hyperbolic motions. The "hyperbolic lines" in "Q" are rays from the origin or petal-shaped curves leaving and re-entering the origin. The left-right shift in "HP" corresponds to a squeeze 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 lawall suggest looking carefully at the quadrant. For example, in thermodynamics the isothermal 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 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:The unit currency sets $x\; =\; 1$. The price currency corresponds to $y$. For

:

we find $u\; >\; 0$, a positive hyperbolic angle. For a "fluctuation" take a new price

:.

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 a basket of consumer goods.
* Quantification of change in marketshare in duopoly.
* Corporate stock splits versus stock buy-back.