In the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
, as in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
of the plane is −1.
Sinh,
cosh and
tanh are
hyperbolic functions.
Polar coordinate system
The polar coordinate system is a
two-dimensional coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in which each
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
on a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
is determined by a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from a reference point and an
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
from a reference direction.
The reference point (analogous to the origin of a
Cartesian system) is called the ''pole'', and the
ray
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (g ...
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 In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigono ...
, we get that the distance between two points given in polar coordinates is
:
The corresponding metric tensor field is:
The straight lines are described by equations of the form
:
where ''r''
0 and θ
0 are the coordinates of the nearest point on the line to the pole.
Quadrant model system
The
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
Equivalently the Poincaré ha ...
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 set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
and the
hyperbolic angle 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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
) point (the
Origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
) 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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, 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 point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' ...
s have axial coordinates, but not every pair of real numbers corresponds to a point.
If
then
is an ideal point.
If
then
is not a point at all.
The distance of a point
to the ''x''-axis is
. To the ''y''-axis it is
.
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
:
:
:
:
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).
''y''
''ℓ'' is the distance along the perpendicular of the given point to its foot (positive on one side and negative on the other).
:
.
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 ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
one of the discoverers of
hyperbolic geometry.
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
:
This formula can be derived from the formulas about
hyperbolic triangles.
The corresponding metric tensor is:
.
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
:
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
:
:
:
:
Horocycle-based coordinate system
Another coordinate system uses the distance from the point to the
horocycle
In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...
through the origin centered around
and the arclength along this horocycle.
Draw the
horocycle
In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...
''h''
O through the origin centered at the
ideal point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' ...
at the end of the ''x''-axis.
From point P draw the line ''p'' asymptotic to the ''x''-axis to the right
ideal point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space.
Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' ...
. ''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
, negative if ''P''
h is between P and
.
The coordinate ''y''
h is the arclength along horocycle ''h''
O from the origin to ''P''
h.
The distance between two points given in these coordinates is
:
The corresponding metric tensor is:
The straight lines are described by equations of the form ''y'' = a constant or
:
where ''x''
0 and ''y''
0 are the coordinates of the point on the line nearest to the ideal point
(i.e. having the largest value of ''x'' on the line).
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 and the origin is mapped to the centre of the boundary circle.
The following equations hold:
:
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
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 th ...
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:
:
Weierstrass coordinates
The Weierstrass coordinates of a point are the Euclidean 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 hyperboloid ...
of the hyperbolic plane, the ''x''-axis is mapped to the (half)
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, cal ...
and the origin is mapped to the point (0,0,1).
The point P with axial coordinates (''x''
''a'', ''y''
''a'') is mapped to
:
Others
Gyrovector coordinates
Gyrovector space
A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
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.
Hyperbolic Triangle Centers: The Special Relativistic Approach
Abraham Ungar, Springer, 2010
, Abraham Ungar, World Scientific, 2010
References
{{reflist
Hyperbolic geometry
Coordinate systems