Vector algebra

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Hyperbolic coordinates plotted on the Cartesian plane: u in blue and v in red.

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

$\{(x,y):x>0,y>0\}=Q$.

Hyperbolic coordinates take values in the hyperbolic plane defined as:

$HP=\{(u,v):u\in \mathbb{ℝ},v>0\}$.

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

For $(x,y)$ in $Q$ take

$u=-\frac{1}{2}\ln \left(\frac{y}{x}\right)$

and

$v=\sqrt{xy}$.

Sometimes the parameter $u$ is called hyperbolic angle and v the geometric mean.

The inverse mapping is

$x=v{e}^u,y=v{e}^{-u}$.

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

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 squeeze mapping applied to Q. Note that hyperbolas in Q do not represent geodesics in this model.

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 as the quadrant model of the hyperbolic plane. 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 irrelevant to the quadrant model.

Applications in physical science

Physical unit relations like:

• V = I R  : Ohm's law
• P = V I  : Electrical power
• P V = k T  : Ideal gas law
• f λ = c : Sine waves

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. Similarly, an isobaric process may trace a hyperbola in the quadrant of absolute temperature and gas density.

For hyperbolic coordinates in the Theory of relativity see the History section below.

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 $x=1$. 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:

$\mathrm{\Delta }u=\frac{1}{2}\log \left(\frac{y}{z}\right)$.

Quantifying exchange rate fluctuation through hyperbolic angle provides an objective, symmetric, and consistent measure. The quantity $\mathrm{\Delta }u$ 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.

History

While the geometric mean is an ancient concept, the hyperbolic angle is contemporary with the development of logarithm, the latter part of the seventeenth century. Gregoire de Saint-Vincent, Marin Mersenne, and Alphonse Antonio de Sarasa evaluated the quadrature of the hyperbola as a function having properties now familiar for the logarithm. The exponential function, the hyperbolic sine, and the hyperbolic cosine followed. As complex function theory referred to infinite series the circular functions sine and cosine seemed to absorb the hyperbolic sine and cosine as depending on an imaginary variable. In the nineteenth century biquaternions came into use and exposed the alternative complex plane called split-complex numbers where the hyperbolic angle is raised to a level equal to the classical angle. In English literature biquaternions were used to model spacetime and show its symmetries. There the hyperbolic angle parameter came to be called rapidity. For relativists, examining the quadrant as the possible future between oppositely directed photons, the geometric mean parameter is temporal.

In 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 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. In tribute to Wolfgang Rindler, the author of the standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called Rindler coordinates.

References

• 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.