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Hyperbolic CoordinatesIn 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.
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