Hyperbolic coordinates

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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

$u = -\frac{1}{2} \log \left( \frac{y}{x} \right)$

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.

1 Quadrant model of hyperbolic geometry
2 Applications in physical science
3 Statistical applications
4 Economic applications

[edit] 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.

[edit] 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, 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.

[edit] Statistical 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.

[edit] 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

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:

$\Delta u = \frac{1}{2} \log \left( \frac{y}{z} \right)$ .

Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity $Δ u$ 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.

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