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Hyperbolic coordinates plotted on the Cartesian plane: u in blue and v in red.

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.

Contents

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:

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.

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.

Economic applications

There are many natural applications of hyperbolic coordinates in economics:

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 Δu is the length of the left-right shift in the hyperbolic motion view of the currency fluctuation.

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