Hyperbolic coordinates
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.
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.
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.
- 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