User:WillemienH/draft Coordinates on the hyperbolic plane

User:WillemienH/draft Coordinates on the hyperbolic plane

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

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

Polar coordinate system


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 around 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 the hyperbolic functions sinus hyperbolicus, cosinus hyperbolicus and tangens hyperbolicus.

Axial coordinates
Axial coordinates xa and ya are found by construcing an y-axis. perpendicular to the x-axis through the Origin.

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 $$ P(x_a, y_a) \text{ to the x-axis is } \operatorname {artanh} \left( \tanh(y_a) \cosh(x_a) \right) $$

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

Lobachevsky coordinates
The Lobachevski coordinates xl and yl are found by dropping a perpendicular onto the x-axis. xl 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)

yl 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 and area between lines and curves.{dubiuos}

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.

Draw the horocycle hO 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 $$ \Omega $$. Ph is the intersection of line p and horocycle hO.

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

The coordinate yh is the arclength along horocycle hO from the origin to Ph.

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.

The following equations hold:

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

Poincare coordinates
The Poincare coordinates of a point are the Euclidean coordinates of the point when the point is mapped in the Poincare disk 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.

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 curve (x,0,sqrt-- ) and the origin is mapped to the point (0,0,1).

The point P with axial coordinates (xa, ya) is mapped to

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

Gyrovector coordinates
Gyrovector space

Hyperbolic Barycentric Coordinates
From Gyrovector space

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.