Hyperbolic coordinates - Definition and Overview

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

exp(u)v = x , exp(−u)v = y.

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

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.

Sample application: 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.