Hyperbolic coordinates

From Wikipedia, the free encyclopedia

In mathematics, hyperbolic coordinates are a useful method of locating points in Quadrant I of the Cartesian plane

<math>\{(x, y) \ :\ x > 0,\ y > 0\ \} = Q\ \!</math >.

Hyperbolic coordinates take values in

<math>HP = \{(u, v) : u \in \mathbb{R}, v > 0 \}</math>.

For <math>(x,y)</math> in <math>Q</math> take

<math>u = -\frac{1}{2} \log \left( \frac{y}{x} \right)</math>

and

<math>v = \sqrt{xy}</math>.

Sometimes the parameter <math>u</math> is called hyperbolic angle and <math>v</math> the geometric mean.

The inverse mapping is

<math>x = v e^u ,\quad y = v e^{-u}</math>.

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

Contents 1 Quadrant model of hyperbolic geometry 2 Applications in physical science 3 Statistical applications 4 Economic applications

[edit] Quadrant model of hyperbolic geometry

The correspondence

<math>Q \leftrightarrow HP</math>

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.

[edit] Applications in physical science

Physical unit relations like:

• E = IR  : Ohm's law
• P = EI  : Electrical power
• PV = kT  : Ideal gas law

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.

[edit] 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 elected representation of regions in a representative democracy begins with selection of a standard for comparison: a particular represented group, whose magnitude and slate magnitude (of representatives) stands at (1,1) in the quadrant.

[edit] Economic applications

There are many natural applications of hyperbolic coordinates in economics:

• Analysis of currency exchange rate fluctuation:

The unit currency sets <math>x = 1</math>. The price currency corresponds to <math>y</math>. For

<math>0 < y < 1</math>

we find <math>u > 0</math>, a positive hyperbolic angle. For a fluctuation take a new price

<math>0 < z < y</math>.

Then the change in u is:

<math>\Delta u = \frac{1}{2} \log \left( \frac{y}{z} \right)</math>.

Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity <math>\Delta u</math> 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.

Image:Question book-3.svg

This article does not cite any references or sources. (January 2008) Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.