Coordinate Systems For The Hyperbolic Plane
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In the hyperbolic plane, as in the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, each point can be uniquely identified by two
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane. In the descriptions below the constant
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
of the plane is −1. Sinh, cosh and tanh are
hyperbolic functions In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
.


Polar coordinate system

The polar coordinate system is a
two-dimensional A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimension ...
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
in which each point on a plane is determined by a
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
from a reference point and an
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
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''. From the hyperbolic law of cosines, we get that the distance between two points given in polar coordinates is :\operatorname (\langle r_1, \theta_1 \rangle, \langle r_2, \theta_2 \rangle) = \operatorname \, \left( \cosh r_1 \cosh r_2 - \sinh r_1 \sinh r_2 \cos (\theta_2 - \theta_1) \right) \,. Let r=r_1=r_2,\theta=\theta_2 - \theta_1, differentiating at \frac=0: :\begin\left.\frac\operatorname (\langle r, \theta_1 \rangle, \langle r, \theta_1+\theta \rangle)\_&=\left.\frac\operatorname \, \left( \cosh^2 r - \sinh^2 r \cos (\theta) \right)\_\\ &=\sinh(r)\end we get the corresponding
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
: (\mathrm s)^2 = (\mathrm r)^2 + \sinh^2 r \, (\mathrm \theta)^2 \,. The straight lines are described by equations of the form : \theta = \theta_0 \pm \frac \quad \text \quad \tanh r = \tanh r_0 \sec (\theta - \theta_0) where ''r''0 and θ0 are the coordinates of the nearest point on the line to the pole.


Quadrant model system

The Poincaré half-plane model is closely related to a model of the hyperbolic plane in the quadrant ''Q'' = . For such a point the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri ...
v = \sqrt and the hyperbolic angle u = \ln \sqrt produce a point (''u,v'') in the upper half-plane. The hyperbolic metric in the quadrant depends on the Poincaré half-plane metric. The motions of the Poincaré model carry over to the quadrant; in particular the left or right shifts of the real axis correspond to hyperbolic rotations of the quadrant. Due to the study of ratios in physics and economics where the quadrant is the universe of discourse, its points are said to be located by hyperbolic coordinates.


Cartesian-style coordinate systems

In hyperbolic geometry rectangles do not exist. The sum of the angles of a quadrilateral in hyperbolic geometry is always less than 4
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s (see Lambert quadrilateral). Also in hyperbolic geometry there 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.


Axial coordinates

Axial coordinates ''x''''a'' and ''y''''a'' are found by constructing a ''y''-axis perpendicular to the ''x''-axis through the origin. Like in the
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, the coordinates are found by dropping perpendiculars from the point onto the ''x'' and ''y''-axes. ''x''''a'' is the distance from the foot of the perpendicular on the ''x''-axis to the origin (regarded as positive on one side and negative on the other); ''y''''a'' is the distance from the foot of the perpendicular on the ''y''-axis to the origin. Every point and most ideal points have axial coordinates, but not every pair of real numbers corresponds to a point. If \tanh^2 (x_a) + \tanh^2 (y_a) = 1 then P(x_a , y_a) is an ideal point. If \tanh^2 (x_a) + \tanh^2 (y_a) > 1 then P(x_a , y_a) is not a point at all. The distance of a point P(x_a , y_a) to the ''x''-axis is \operatorname \left( \tanh(y_a) \cosh(x_a) \right) . To the ''y''-axis it is \operatorname \left( \tanh(x_a) \cosh(y_a) \right) . The relationship of axial coordinates to polar coordinates (assuming the origin is the pole and that the positive ''x''-axis is the polar axis) is : x = \operatorname \, (\tanh r \cos \theta) : y = \operatorname \, (\tanh r \sin \theta) : r = \operatorname \, (\sqrt \, ) : \theta = 2 \operatorname \, \left( \frac \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, the same as in
axial coordinates Axial may refer to: * one of the anatomical directions describing relationships in an animal body * In geometry: :* a geometric term of location :* an axis of rotation * In chemistry, referring to an axial bond * a type of modal frame, in music ...
). ''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) = \tanh(y_a) \cosh(x_a) \ , \ \tanh(y_a) = \frac . The Lobachevsky coordinates are useful for integration for length of curves and area between lines and curves. Lobachevsky coordinates are named after
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky (; , ; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, kno ...
, one of the discoverers of
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
. Construct a Cartesian-like coordinate system as follows. Choose a line (the ''x''-axis) in the hyperbolic plane (with a standardized curvature of −1) and label the points on it by their distance from an origin (''x''=0) point on the ''x''-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates ''x'' and ''y'' by dropping a perpendicular onto the ''x''-axis. ''x'' will be the label of the foot of the perpendicular. ''y'' will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). Then the distance between two such points will be :\operatorname (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname \left( \cosh y_1 \cosh (x_2 - x_1) \cosh y_2 - \sinh y_1 \sinh y_2 \right) \,. This formula can be derived from the formulas about
hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three point (geometry), points called ''angles'' or ''vertices''. Just as in the Euclidea ...
s. The corresponding metric tensor is: (\mathrm s)^2 = \cosh^2 y \, (\mathrm x)^2 + (\mathrm y)^2 . In this coordinate system, straight lines are either perpendicular to the ''x''-axis (with equation ''x'' = a constant) or described by equations of the form : \tanh y = A \cosh x + B \sinh x \quad \text \quad A^2 < 1 + B^2 where ''A'' and ''B'' are real parameters which characterize the straight line. The relationship of Lobachevsky coordinates to polar coordinates (assuming the origin is the pole and that the positive ''x''-axis is the polar axis) is : x = \operatorname \, (\tanh r \cos \theta) : y = \operatorname \, (\sinh r \sin \theta) : r = \operatorname \, (\cosh x \cosh y) : \theta = 2 \operatorname \, \left( \frac \right) \,.


Horocycle-based coordinate system

Another coordinate system represents each hyperbolic point P by two real numbers, defined relative to some given horocycle. These numbers are the hyperbolic distance x_h from P to the horocycle, and the (signed)
arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
y_h along the horocycle between a fixed reference point O and P_h, where P_h is the closest point on the horocycle to P.


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
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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 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)


Poincaré coordinates

The Poincaré coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
of the hyperbolic plane, the ''x''-axis is mapped to the segment and the origin is mapped to the centre of the boundary circle. The Poincaré coordinates, in terms of the Beltrami coordinates, are: : x_p = \frac, \ \ y_p = \frac


Weierstrass coordinates

The Weierstrass coordinates of a point are the Cartesian coordinates of the point when the point is mapped in the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloi ...
of the hyperbolic plane, the ''x''-axis is mapped to the (half)
hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...
(t \ , \ 0 \ , \ \sqrt ) and the origin is mapped to the point (0,0,1). The point P with axial coordinates (''x''''a'', ''y''''a'') is mapped to : \left( \frac \ , \ \frac \ , \ \frac \right)


Others


Gyrovector coordinates

Gyrovector space


Hyperbolic barycentric coordinates

From Gyrovector space#Triangle centers 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.Hyperbolic Triangle Centers: The Special Relativistic Approach
Abraham Ungar, Springer, 2010

, Abraham Ungar, World Scientific, 2010


References

{{reflist Hyperbolic geometry Coordinate systems