Coordinate systems for the hyperbolic plane

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. In Euclidean geometry, all Cartesian coordinate systems are fundamentally the same, but hyperbolic geometry there are fundamental differences.

This article tries to give an overview of fundamental different coordinate systems for the 2-dimensional hyperbolic plane.

Polar coordinate system

Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, or polar angle

Cartesian style coordinate systems

In hyperbolic geometry rectangles do not exist. (Lambert quadrilateral, the sum of the angles of a quadrilateral in hyperbolic geometry is always less than 4 right angles) also in hyperbolic geometrythere are no equidistant lines. (see hypercycles). This all has influences on the coordinate systems.

There are however different coordinate systems for hyperbolic plane geometry. All are based on choosing a real (non ideal) point (the Origin) on a chosen directed line (the x-axis) and after that many choices exist.

In the descriptions below the constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions.

Axial coordinates

Axial coordinates x_a and y_a are found by construcing an y-axis. perpendicular to the x-axis through the Origin.^[1]

Like in the Cartesian coordinate system the coordinates are found by Dropping a perpendiculars from the point onto the x and y'-axis. xa is the distance from the foot of the perpendicular to the x-axis to the origin.(positive on one side and negative on the other) ya is the distance from the foot of the perpendicular to the y-axis to the origin.

Every point and most ideal points have axial coordinates, but not every pair of real numbers give a point. If $\tanh(x_a)^2 + \tanh(y_a)^2 = 1 \text{ then } P(x_a , y_a)$ is an ideal point.

If $\tanh(x_a)^2 + \tanh(y_a)^2 > 1 \text{ then } P(x_a , y_a)$ is not a point at all.

The distance sx of a point $P(x_a , y_a) \text{ to the }x\text{-axis is } \operatorname {artanh} \left( \tanh(y_a) \cosh(x_a) \right)$

And to the y-axis it is. $\operatorname {artanh} \left( \tanh(x_a) \cosh(y_a) \right)$

Lobachevsky coordinates

The Lobachevsky coordinates x_ℓ and y_ℓ are found by dropping a perpendicular onto the x-axis. x_ℓ is the distance from the foot of the perpendicular to the x-axis to the origin.(positive on one side and negative on the other, same as in axial coordinates)^[1]

y_ℓ is the distance along the perpendicular of the given point to its foot (positive on one side and negative on the other).

$x_l = x_a \ , \ \tanh(y_l) = \frac{\tanh(y_a)}{\cosh(x_a)}$

The Lobachevsky coordinates are useful for integration for length of curves^[2] and area between lines and curves.

Lobachevsky coordinates are named after Nikolai Lobachevsky one of the discoverers of hyperbolic geometry

Horocycle-based coordinate system

Another coordinate system uses the distance from the point to the horocycle through the Origin centered around $\Omega {{=}} (0, + \infty )$ and the arclength along this horocycle.^[3]

Draw the horocycle h_O through the origin centered at the ideal point $\Omega$ at the end of the 'x-axis.

From point P draw the line p asymptopic to the x-axis to the right ideal point $\Omega$. P_h is the intersection of line p and horocycle h_O.

The coordinate x_h is the distance from P to P_h positive if P is between P_h and $\Omega$, negative if P_h is between P and $\Omega$

The coordinate y_h is the arclength along horocycle h_O from the origin to P_h.

Model-based coordinate systems

Model-based coordinate systems use one of the models of hyperbolic geometry and take the Euclidean coordinates inside the model as the hyperbolic coordinates.

Beltrami coordinates

The Beltrami coordinates of a point are the Euclidean coordinates of the point when the point is mapped in the Beltrami–Klein model of the hyperbolic plane, the x-axis is mapped to the segment (−1,0) − (1,0) and the origin is mapped to the centre of the boundary circle.^[1]

The following equations hold:

$x_b = \tanh (x_a), \ y_b = \tanh(y_a)$

Poincaré coordinates

The Poincaré coordinates of a point are the Euclidean coordinates of the point when the point is mapped in the Poincaré disk model of the hyperbolic plane,^[1] the x-axis is mapped to the segment (−1,0) − (1,0) and the origin is mapped to the centre of the boundary circle.

Weierstrass coordinates

The Weierstrass coordinates of a point are the Euclidean coordinates of the point when the point is mapped in the hyperboloid model of the hyperbolic plane, the x-axis is mapped to the (half) hyperbola $(t \ , \ 0 \ , \ \sqrt{t^2+1} )$ and the origin is mapped to the point (0,0,1).^[1]

The point P with axial coordinates (x_a, y_a) is mapped to

$\left( \frac{\tanh x_a}{\sqrt{1 -\tanh^2 x_a - \tanh^2 y_a }} \ , \ \frac{\tanh y_a}{\sqrt{1 -\tanh^2 x_a - \tanh^2 y_a }} \ , \ \frac{1}{\sqrt{1 -\tanh^2 x_a - \tanh^2 y_a }} \right)$

Others

Gyrovector coordinates

Gyrovector space

Hyperbolic barycentric coordinates

From Gyrovector space#triangle center

The study of triangle centers traditionally is concerned with Euclidean geometry, but triangle centers can also be studied in hyperbolic geometry. Using gyrotrigonometry, expressions for trigonometric barycentric coordinates can be calculated that have the same form for both euclidean and hyperbolic geometry. In order for the expressions to coincide, the expressions must not encapsulate the specification of the anglesum being 180 degrees.^[4][5][6]

References

1. Martin, George E. (1998). The foundations of geometry and the non-Euclidean plane (Corrected 4. print. ed.). New York, NY: Springer. pp. 447–450. ISBN 0387906940.
2. Smorgorzhevsky, A.S. (1982). Lobachevskian geometry. Moscow: Mir. pp. 64–68.
3. Ramsay, Arlan; Richtmyer, Robert D. (1995). Introduction to hyperbolic geometry. New York: Springer-Verlag. pp. 97–103. ISBN 0387943390.
4. Hyperbolic Barycentric Coordinates, Abraham A. Ungar, The Australian Journal of Mathematical Analysis and Applications, AJMAA, Volume 6, Issue 1, Article 18, pp. 1–35, 2009
5. Hyperbolic Triangle Centers: The Special Relativistic Approach, Abraham Ungar, Springer, 2010
6. Barycentric Calculus In Euclidean And Hyperbolic Geometry: A Comparative Introduction, Abraham Ungar, World Scientific, 2010