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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
:
ight)
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

Statistical applications

Economic applications

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 "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
:
we find $u\; >\; 0$, a positive hyperbolic angle. For a ''fluctuation'' take a new price
:.
Then the change in ''u'' is:
:
ight).
Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity $Delta\; u$ 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.