In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane
Hyperbolic coordinates take values in
For (x,y) in Q take
and
Sometimes the parameter u is called hyperbolic angle and v the geometric mean.
The inverse mapping is
This is a continuous mapping, but not an analytic function.
Contents 
The correspondence
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 petalshaped curves leaving and reentering the origin. The leftright shift in HP corresponds to a squeeze mapping applied to Q.
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.
There are many natural applications of hyperbolic coordinates in economics:
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:
Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity Δu is the length of the leftright shift in the hyperbolic motion view of the currency fluctuation.
