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.
Since HP carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence
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.
Insight from the metric space HP shows that the open set Q has only the origin as boundary when viewed through the correspondence.
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 hyperbolic coordinates in the theory of relativity see the History section.
Evidently the diagonals divide the rhombus into four congruent right triangles.
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 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.
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).
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.
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 1875 Johann von Thünen published a theory of natural wages[1] which used geometric mean of a subsistence wage and market value of the labor using the employer's capital.
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[2] 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.
[3] In tribute to Wolfgang Rindler, the author of a standard introductory university-level textbook on relativity, hyperbolic coordinates of spacetime are called Rindler coordinates.