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 ∈ R, v > 0 }.

For (x,y) in Q take

u = −1/2 log(y/x)

and

v = √(xy).

Sometimes the parameter u is called hyperbolic angle and v the geometric mean.

The inverse mapping is

x = v e^u , y = v e^−u .

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

[edit] Quadrant model of hyperbolic geometry

The correspondence

Q ↔ 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 "hyperbolic rotation" in Q.

[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 representation of regions in a democracy begins with selection of a standard for comparison, a particular represented group, whose magnitude and slate 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:

Δu = (1/2)log(y/z).

Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure.The quantity Δu is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.