# Hyperbolic coordinates

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

- .

Hyperbolic coordinates take values in the hyperbolic plane defined as:

- .

These coordinates in *HP* are useful for studying logarithmic comparisons of direct proportion in *Q* and measuring deviations from direct proportion.

For in take

and

- .

The parameter *u* is the hyperbolic angle to (*x, y*) and *v* is the geometric mean of *x* and *y*.

The inverse mapping is

- .

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

## Alternative quadrant metric

Since *HP* carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence
brings this structure to *Q*. It can be grasped using the notion of hyperbolic motions. Since geodesics in *HP* are semicircles with centers on the boundary, the geodesics in *Q* are obtained from the correspondence and turn out to be rays from the origin or petal-shaped curves leaving and re-entering the origin. And the hyperbolic motion of *HP* given by a left-right shift corresponds to a squeeze mapping applied to *Q*.

Since hyperbolas in *Q* correspond to lines parallel to the boundary of *HP*, they are horocycles in the metric geometry of *Q*.

If one only considers the Euclidean topology of the plane and the topology inherited by *Q*, then the lines bounding *Q* seem close to *Q*. Insight from the metric space *HP* shows that the open set *Q* has only the origin as boundary when viewed through the correspondence. Indeed, consider rays from the origin in *Q*, and their images, vertical rays from the boundary *R* of *HP*. Any point in *HP* is an infinite distance from the point *p* at the foot of the perpendicular to *R*, but a sequence of points on this perpendicular may tend in the direction of *p*. The corresponding sequence in *Q* tends along a ray toward the origin. The old Euclidean boundary of *Q* is no longer relevant.

## Applications in physical science

Fundamental physical variables are sometimes related by equations of the form *k* = *x y*. For instance, *V* = *I R* (Ohm's law), *P* = *V I* (electrical power), *P V* = *k T* (ideal gas law), and *f* λ = *v* (relation of wavelength, frequency, and velocity in the wave medium). When the *k* is constant, the other variables lie on a hyperbola, which is a horocycle in the appropriate *Q* quadrant.

For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass *M* of gas with changing volume will have variable density δ = *M / V*, and the ideal gas law may be written *P = k T* δ so that an isobaric process traces a hyperbola in the quadrant of absolute temperature and gas density.

For hyperbolic coordinates in the theory of relativity see the History section.

## 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.

## Economic applications

There are many natural applications of hyperbolic coordinates in economics:

- Analysis of currency exchange rate fluctuation:

The unit currency sets . The price currency corresponds to . For

we find , 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 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.

## History

The geometric mean is an ancient concept, but hyperbolic angle was developed in this configuration by Gregoire de Saint-Vincent. He was attempting to perform quadrature with respect to the rectangular hyperbola *y* = 1/*x*. That challenge was a standing open problem since Archimedes performed the quadrature of the parabola. The curve passes through (1,1) where it is opposite the origin (mathematics) in a unit square. The other points on the curve can be viewed as rectangles having the same area as this square. Such a rectangle may be obtained by applying a squeeze mapping to the square. Another way to view these mappings is via hyperbolic sectors. Starting from (1,1) the hyperbolic sector of unit area ends at (e, 1/e), where e is 2.71828…, according to the development of Leonhard Euler in *Introduction to the Analysis of the Infinite* (1748).

Taking (e, 1/e) as the vertex of rectangle of unit area, and applying again the squeeze that made it from the unit square, yields Generally n squeezes yields A. A. de Sarasa noted a similar observation of G. de Saint Vincent, that as the abscissas increased in a geometric series, the sum of the areas against the hyperbola increased in arithmetic series, and this property corresponded to the **logarithm** already in use to reduce multiplications to additions. Euler’s work made the natural logarithm a standard mathematical tool, and elevated mathematics to the realm of transcendental functions. The hyperbolic coordinates are formed on the original picture of G. de Saint-Vincent, which provided the quadrature of the hyperbola, and transcended the limits of algebraic functions.

In special relativity the focus is on the 3-dimensional hypersurface in the future of spacetime where various velocities arrive after a given proper time. Scott Walter[1] explains that in November 1907 Hermann Minkowski alluded to a well-known three-dimensional hyperbolic geometry while speaking to the Göttingen Mathematical Society, but not to a four-dimensional one.[2] In tribute to Wolfgang Rindler, the author of a standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called Rindler coordinates.

## References

- Walter (1999) page 6
- Walter (1999) page 8

- David Betounes (2001)
*Differential Equations: Theory and Applications*, page 254, Springer-TELOS, ISBN 0-387-95140-7 . - Scott Walter (1999). "The non-Euclidean style of Minkowskian relativity". Chapter 4 in: Jeremy J. Gray (ed.),
*The Symbolic Universe: Geometry and Physics 1890-1930*, pp. 91–127. Oxford University Press. ISBN 0-19-850088-2.