In

^{2}. As a consequence, all hyperbolic triangles have an area that is less than or equal to ''R''^{2}π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.
As in Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

, each hyperbolic triangle has an

Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line.
However, in hyperbolic geometry, a regular apeirogon has sides of any length (i.e., it remains a polygon).
The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel (see lines above).
If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric 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 horos ...

s.
If the bisectors are diverging parallel then a pseudogon (distinctly different from an apeirogon) can be inscribed in hypercycles (all vertices are the same distance of a line, the axis, also the midpoint of the side segments are all equidistant to the same axis.)

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 . ...

, it is usual to assume a scale in which the curvature ''K'' is −1.
This results in some formulas becoming simpler. Some examples are:
* The area of a triangle is equal to its angle defect in 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 horos ...

so that a line that is tangent at one endpoint is limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...

to the radius through the other endpoint has a length of 1.
* The ratio of the arc lengths between two radii of two concentric 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 horos ...

s where the horocycles are a distance 1 apart is ''e'' :1.

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 horos ...

through the origin centered around $(0,\; +\; \backslash infty\; )$ and the length along this horocycle.
Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

_{0}, ''y''_{0}, ''A'', and ''α'' are parameters):
ultraparallel to the ''x''-axis
:$\backslash tanh\; (y)\; =\; \backslash tanh\; (y\_0)\; \backslash cosh\; (x\; -\; x\_0)$
asymptotically parallel on the negative side
:$\backslash tanh\; (y)\; =\; A\; \backslash exp\; (x)$
asymptotically parallel on the positive side
:$\backslash tanh\; (y)\; =\; A\; \backslash exp\; (-\; x)$
intersecting perpendicularly
:$x\; =\; x\_0$
intersecting at an angle ''α''
:$\backslash tanh\; (y)\; =\; \backslash tan\; (\backslash alpha)\; \backslash sinh\; (x\; -\; x\_0)$
Generally, these equations will only hold in a bounded domain (of ''x'' values). At the edge of that domain, the value of ''y'' blows up to ±infinity. See also Coordinate systems for the hyperbolic plane#Polar coordinate system.

parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were

Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.
It is said that

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 . ...

) in a three-dimensional Euclidean space.
Other useful

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 . ...

, which is indifferent to the coordinate chart used. 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 ...

(horolation) — two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom.
** translation along a straight line — two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degrees of freedom.
* Orientation reversing
** reflection through a line — one reflection; two degrees of freedom.
** combined reflection through a line and translation along the same line — the reflection and translation commute; three reflections required; three degrees of freedom.

^{1,''n''}, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic ''n''-space. The stabilizer of any particular line is isomorphic to the

Hyperbolic geometry: The first 150 years

', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24. * Reynolds, William F., (1993) ''Hyperbolic Geometry on a Hyperboloid'',

Hyperbolic Geometry

', MSRI Publications, volume 31.

University of New Mexico * ttps://www.youtube.com/watch?v=B16YjC9OS0k&mode=user&search= "The Hyperbolic Geometry Song"A short music video about the basics of Hyperbolic Geometry available at YouTube. * * *

More on hyperbolic geometry, including movies and equations for conversion between the different models

University of Illinois at Urbana-Champaign

Hyperbolic Voronoi diagrams made easy, Frank Nielsen

* , interactive instructional website.

Hyperbolic Planar Tesselations

Models of the Hyperbolic Plane

{{DEFAULTSORT:Hyperbolic Geometry

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...

. The parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

of Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

is replaced with:
:For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.
(Compare the above with Playfair's axiom
In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
''In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through th ...

, the modern version of Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

's parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

.)
Hyperbolic plane geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

is also the geometry of pseudospherical surfaces, surfaces with a constant negative 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 . ...

. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...

resemble the hyperbolic plane.
A modern use of hyperbolic geometry is in the theory of special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

, particularly the Minkowski model.
When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and gro ...

finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

(spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

), parabolic geometry (Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

), and hyperbolic geometry.
In the former Soviet Union
The post-Soviet states, also known as the former Soviet Union (FSU), the former Soviet Republics and in Russia as the near abroad (russian: links=no, ближнее зарубежье, blizhneye zarubezhye), are the 15 sovereign states that we ...

, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.
This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. T ...

for more information on hyperbolic geometry extended to three and more dimensions.
Properties

Relation to Euclidean geometry

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the onlyaxiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

atic difference is the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

.
When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not s ...

.
There are two kinds of absolute geometry, Euclidean and hyperbolic.
All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's ''Elements'', are valid in Euclidean and hyperbolic geometry.
Propositions 27 and 28 of Book One of Euclid's ''Elements'' prove the existence of parallel/non-intersecting lines.
This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry; new concepts need to be introduced.
Further, because of the angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle an ...

, hyperbolic geometry has an absolute scale
There is no single definition of an absolute scale. In statistics and measurement theory, it is simply a ratio scale in which the unit of measurement is fixed, and values are obtained by counting. According to another definition it is a system of ...

, a relation between distance and angle measurements.
Lines

Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended. Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary. When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines. These properties are all independent of themodel
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models ...

used, even if the lines may look radically different.
Non-intersecting / parallel lines

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines inEuclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axio ...

:
:For any line ''R'' and any point ''P'' which does not lie on ''R'', in the plane containing line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.
This implies that there are through ''P'' an infinite number of coplanar lines that do not intersect ''R''.
These non-intersecting lines are divided into two classes:
* Two of the lines (''x'' and ''y'' in the diagram) are limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...

s (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of 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 ...

s at the "ends" of ''R'', asymptotically approaching ''R'', always getting closer to ''R'', but never meeting it.
* All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ''ultraparallel'', ''diverging parallel'' or sometimes ''non-intersecting.''
Some geometers simply use the phrase "''parallel'' lines" to mean "''limiting parallel'' lines", with ''ultraparallel'' lines meaning just ''non-intersecting''.
These limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...

s make an angle ''θ'' with ''PB''; this angle depends only on the 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 . ...

of the plane and the distance ''PB'' and is called the angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle an ...

.
For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.
Circles and disks

In hyperbolic geometry, the circumference of a circle of radius ''r'' is greater than $2\; \backslash pi\; r$. Let $R\; =\; \backslash frac$, where $K$ is theGaussian 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 . ...

of the plane. In hyperbolic geometry, $K$ is negative, so the square root is of a positive number.
Then the circumference of a circle of radius ''r'' is equal to:
:$2\backslash pi\; R\; \backslash sinh\; \backslash frac\; \backslash ,.$
And the area of the enclosed disk is:
:$4\backslash pi\; R^2\; \backslash sinh^2\; \backslash frac\; =\; 2\backslash pi\; R^2\; \backslash left(\backslash cosh\; \backslash frac\; -\; 1\backslash right)\; \backslash ,.$
Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than $2\backslash pi$, though it can be made arbitrarily close by selecting a small enough circle.
If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius ''r'' is: $\backslash frac$
Hypercycles and horocycles

In hyperbolic geometry, there is no line all of whose points are equidistant from another line. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle. Another special curve is thehorocycle
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 horos ...

, a curve whose normal radii (perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...

lines) are all limiting parallel
In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line l through a point P not on line R; however, in the plane, two parallels may be closer to l than all others (one in each direction of R) ...

to each other (all converge asymptotically in one direction to the same 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 ...

, the centre of the horocycle).
Through every pair of points there are two horocycles. The centres of the horocycles are 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 ...

s of the perpendicular bisector of the line-segment between them.
Given any three distinct points, they all lie on either a line, hypercycle, 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 horos ...

, or circle.
The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points. The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger than the arc-length of a horocycle connecting two points.
If the Gaussian curvature of the plane is −1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1.
Triangles

Unlike Euclidean triangles, where the angles always add up to πradian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

s (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

s (180°, a straight angle). The difference is referred to as the defect
A defect is a physical, functional, or aesthetic attribute of a product or service that exhibits that the product or service failed to meet one of the desired specifications. Defect, defects or defected may also refer to:
Examples
* Angular defe ...

.
The area of a hyperbolic triangle is given by its defect in radians multiplied by ''R''incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incente ...

. In hyperbolic geometry, if all three of its vertices lie on a 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 horos ...

or hypercycle, then the triangle has no circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

.
As in spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.
Regular apeirogon

A special polygon in hyperbolic geometry is the regularapeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...

, a uniform polygon with an infinite number of sides.
In Tessellations

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane withregular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a seque ...

s as faces.
There are an infinite number of uniform tilings based on the Schwarz triangles (''p'' ''q'' ''r'') where 1/''p'' + 1/''q'' + 1/''r'' < 1, where ''p'', ''q'', ''r'' are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains.
Standardized Gaussian curvature

Though hyperbolic geometry applies for any surface with a constant negativeradian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

s.
* The area of a horocyclic sector is equal to the length of its horocyclic arc.
* An arc of a Cartesian-like coordinate systems

In hyperbolic geometry, the sum of the angles of aquadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

is always less than 360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems.
There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the ''x''-axis) and after that many choices exist.
The Lobachevski coordinates ''x'' and ''y'' are found 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).
Another coordinate system measures the distance from the point to the Distance

A Cartesian-like coordinate system (''x, y'') on the oriented hyperbolic plane is constructed as follows. Choose a line in the hyperbolic plane together with an orientation and an origin ''o'' on this line. Then: *the ''x''-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin; *the ''y''-coordinate is the signeddistance
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 the point to the line, with the sign according to whether the point is on the positive or negative side of the oriented line.
The distance between two points represented by (''x_i, y_i''), ''i=1,2'' in this coordinate system is
$$\backslash operatorname\; (\backslash langle\; x\_1,\; y\_1\; \backslash rangle,\; \backslash langle\; x\_2,\; y\_2\; \backslash rangle)\; =\; \backslash operatorname\; \backslash left(\; \backslash cosh\; y\_1\; \backslash cosh\; (x\_2\; -\; x\_1)\; \backslash cosh\; y\_2\; -\; \backslash sinh\; y\_1\; \backslash sinh\; y\_2\; \backslash right)\; \backslash ,.$$
This formula can be derived from the formulas about hyperbolic triangles.
The corresponding metric tensor field is: $(\backslash mathrm\; s)^2\; =\; \backslash cosh^2\; y\; \backslash ,\; (\backslash mathrm\; x)^2\; +\; (\backslash mathrm\; y)^2$.
In this coordinate system, straight lines take one of these forms ((''x'', ''y'') is a point on the line; ''x''History

Since the publication ofEuclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...

circa 300 BCE, many geometers
A geometer is a mathematician whose area of study is geometry.
Some notable geometers and their main fields of work, chronologically listed, are:
1000 BCE to 1 BCE
* Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra
* ...

made attempts to prove the Proclus
Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosopher ...

, Ibn al-Haytham
Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...

(Alhacen), Omar Khayyám, Nasīr al-Dīn al-Tūsī, Witelo
Vitello ( pl, Witelon; german: Witelo; – 1280/1314) was a friar, theologian, natural philosopher and an important figure in the history of philosophy in Poland.
Name
Vitello's name varies with some sources. In earlier publications he was qu ...

, Gersonides
Levi ben Gershon (1288 – 20 April 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew by the abbreviation of first letters as ''RaLBaG'', was a medieval French Jewish philosoph ...

, Alfonso
Alphons (Latinized ''Alphonsus'', ''Adelphonsus'', or ''Adefonsus'') is a male given name recorded from the 8th century (Alfonso I of Asturias, r. 739–757) in the Christian successor states of the Visigothic kingdom in the Iberian peninsula. ...

, and later Giovanni Gerolamo Saccheri, John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the roy ...

, Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subje ...

, and Legendre.
Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry.
The theorems of Alhacen, Khayyam and al-Tūsī on quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

s, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.
In the 18th century, Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subje ...

introduced the hyperbolic functions and computed the area of a hyperbolic triangle.
19th-century developments

In the 19th century, hyperbolic geometry was explored extensively by Nikolai Ivanovich Lobachevsky, János Bolyai,Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.
Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...

" causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.
In 1868, Eugenio Beltrami provided models (see below) of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bico ...

Euclidean geometry was.
The term "hyperbolic geometry" was introduced by Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and gro ...

in 1871. Klein followed an initiative of Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...

to use the transformations of projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, p ...

to produce isometries. The idea used a conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spec ...

or quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is ...

to define a region, and used cross ratio to define a metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...

. The projective transformations that leave the conic section or quadric stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...

are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."
For more history, see article on non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...

, and the references Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...

and Milnor.
Philosophical consequences

The discovery of hyperbolic geometry had importantphilosophical
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some s ...

consequences. Before its discovery many philosophers (for example Hobbes
Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book ''Leviathan'', in which he expounds an influent ...

and Spinoza
Baruch (de) Spinoza (born Bento de Espinosa; later as an author and a correspondent ''Benedictus de Spinoza'', anglicized to ''Benedict de Spinoza''; 24 November 1632 – 21 February 1677) was a Dutch philosopher of Portuguese-Jewish origin, ...

) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in ''Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...

''.
Kant
Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...

in the ''Critique of Pure Reason'' came to the conclusion that space (in Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians
Boeotia ( ), sometimes Latinized as Boiotia or Beotia ( el, Βοιωτία; modern: ; ancient: ), formerly known as Cadmeis, is one of the regional units of Greece. It is part of the region of Central Greece. Its capital is Livadeia, and its l ...

", which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians").
The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in mathematical rigour
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...

, analytical philosophy and logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.
Geometry of the universe (spatial dimensions only)

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature? Lobachevsky had already tried to measure the curvature of the universe by measuring theparallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby object ...

of Sirius
Sirius is the brightest star in the night sky. Its name is derived from the Greek word , or , meaning 'glowing' or 'scorching'. The star is designated α Canis Majoris, Latinized to Alpha Canis Majoris, and abbreviated Alpha C ...

and treating Sirius as the ideal point of an angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle an ...

. He realised that his measurements were not precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the absolute length is at least one million times the diameter of the earth's orbit
Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. One complete orbit takes days (1 sidereal year), during which time Earth ...

(, 10 parsec
The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, ...

).
Some argue that his measurements were methodologically flawed.
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

, with his sphere-world
The idea of a sphere-world was constructed by Henri Poincaré who, while pursuing his argument for conventionalism (see philosophy of space and time), offered a thought experiment about a sphere with strange properties.
The concept
Poincaré as ...

thought experiment
A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences.
History
The ancient Greek ''deiknymi'' (), or thought experiment, "was the most ...

, came to the conclusion that everyday experience does not necessarily rule out other geometries.
The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.
Geometry of the universe (special relativity)

Special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

places space and time on equal footing, so that one considers the geometry of a unified spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diff ...

instead of considering space and time separately. Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity
Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his '' Dialogue Concerning the Two Chief World Systems'' using ...

).
In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the in ...

, de Sitter space and anti-de Sitter space
In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872 ...

, corresponding to zero, positive and negative curvature respectively.
Hyperbolic geometry enters special relativity through rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with ...

, which stands in for velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity ...

, and is expressed by a hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functio ...

. The study of this velocity geometry has been called kinematic geometry. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).
Physical realizations of the hyperbolic plane

The hyperbolic plane is a plane where every point is asaddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...

. There exist various pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surfac ...

s in Euclidean space that have a finite area of constant negative Gaussian curvature.
By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...

of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surfac ...

is due to William Thurston.
The art of crochet
Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread, or strands of other materials. The name is derived from the French term ''crochet'', meaning 'hook'. Hooks can be made from a variety o ...

has been used (see ) to demonstrate hyperbolic planes, the first such demonstration having been made by Daina Taimiņa.
In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the " hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling).
Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson, have been made available by Jeff Weeks.
Models of the hyperbolic plane

Variouspseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surfac ...

s – surfaces with constant negative Gaussian curvature – can be embedded in 3-dimensional space under the standard Euclidean metric, and so can be made into tangible physical models. Of these, the tractoid (often called the pseudosphere) is the best known; using the tractoid as a model of the hyperbolic plane is analogous to using a cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or line ...

or cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an inf ...

as a model of the Euclidean plane. However, the entire hyperbolic plane cannot be embedded into Euclidean space in this way, and various other models are more convenient for abstractly exploring hyperbolic geometry.
There are four model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models ...

s commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, 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 Poincar ...

, and the Lorentz or 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 ...

. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry.
Despite their names, the first three mentioned above were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein. All these models are extendable to more dimensions.
The Beltrami–Klein model

The Beltrami–Klein model, also known as the projective disk model, Klein disk model and Klein model, is named after Eugenio Beltrami andFelix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and gro ...

.
For the two dimensions this model uses the interior of the unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

for the complete hyperbolic plane, and the chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ...

of this circle are the hyperbolic lines.
For higher dimensions this model uses the interior of the unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...

, and the chords
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* Chord ...

of this ''n''-ball are the hyperbolic lines.
* This model has the advantage that lines are straight, but the disadvantage that angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...

s are distorted (the mapping is not conformal), and also circles are not represented as circles.
* The distance in this model is half the logarithm of the cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, t ...

, which was introduced by Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...

in projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, p ...

.
The Poincaré disk model

The Poincaré disk model, also known as the conformal disk model, also employs the interior of theunit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

, but lines are represented by arcs of circles that are orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

to the boundary circle, plus diameters of the boundary circle.
* This model preserves angles, and is thereby conformal. All isometries within this model are therefore Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying '' ...

s.
* Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle.
* 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 horos ...

s are circles within the disk which are tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. ...

to the boundary circle, minus the point of contact.
* Hypercycles are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.
The Poincaré half-plane model

ThePoincaré 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 Poincar ...

takes one-half of the Euclidean plane, bounded by a line ''B'' of the plane, to be a model of the hyperbolic plane. The line ''B'' is not included in the model.
The Euclidean plane may be taken to be a plane with 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 i ...

and the x-axis
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 ...

is taken as line ''B'' and the half plane is the upper half (''y'' > 0 ) of this plane.
* Hyperbolic lines are then either half-circles orthogonal to ''B'' or rays perpendicular to ''B''.
* The length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by t ...

$(x,\; y)\; \backslash mapsto\; (\backslash lambda\; x,\; \backslash lambda\; y),\backslash quad\; \backslash lambda\; >\; 0\; .$
* Like the Poincaré disk model, this model preserves angles, and is thus conformal. All isometries within this model are therefore Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying '' ...

s of the plane.
* The half-plane model is the limit of the Poincaré disk model whose boundary is tangent to ''B'' at the same point while the radius of the disk model goes to infinity.
The hyperboloid model

Thehyperboloid 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 ...

or Lorentz model employs a 2-dimensional hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...

of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the in ...

. This model is generally credited to Poincaré, but Reynolds says that Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics is an area of knowledge that includes the topics of nu ...

used this model in 1885
* This model has direct application to special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The la ...

, as Minkowski 3-space is a model for spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diff ...

, suppressing one spatial dimension. One can take the hyperboloid to represent the events (positions in spacetime) that various inertially moving observers, starting from a common event, will reach in a fixed proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...

.
* The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity
In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with ...

between the two corresponding observers.
* The model generalizes directly to an additional dimension: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.
The hemisphere model

Thehemisphere
Hemisphere refers to:
* A half of a sphere
As half of the Earth
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemisphere
** Land and water hemispheres
* A half of the (geocentric) celestia ...

model is not often used as model by itself, but it functions as a useful tool for visualising transformations between the other models.
The hemisphere model uses the upper half of the unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A uni ...

:
$x^2\; +\; y^2\; +z^2\; =\; 1\; ,\; z\; >\; 0.$
The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere.
The hemisphere model is part of a Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...

, and different projections give different models of the hyperbolic plane:
* Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter t ...

from $(0,0,\; -1)$ onto the plane $z\; =\; 0$ projects corresponding points on the Poincaré disk model
* Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter t ...

from $(0,0,\; -1)$ onto the surface $x^2\; +\; y^2\; -\; z^2\; =\; -1\; ,\; z\; >\; 0$ projects corresponding points on 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 ...

* Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter t ...

from $(-1,0,0)$ onto the plane $x=1$ projects corresponding points on 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 Poincar ...

* Orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...

onto a plane $z\; =\; C$ projects corresponding points on the Beltrami–Klein model.
* Central projection from the centre of the sphere onto the plane $z\; =\; 1$ projects corresponding points on the Gans Model
See further: Connection between the models (below)
The Gans model

In 1966 David Gans proposed a flattened hyperboloid model in the journal ''American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an ...

''. It is an orthographic projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogona ...

of the hyperboloid model onto the xy-plane.
This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.
* Unlike the Klein or the Poincaré models, this model utilizes the entire 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 ...

.
* The lines in this model are represented as branches of a 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, ca ...

.
The band model

The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by $\backslash $, the metric is given by $,\; dz,\; \backslash sec\; (\backslash operatorname\; z)$.Connection between the models

All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the samemetric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

, namely the hyperbolic plane.
The characteristic feature of the hyperbolic plane itself is that it has a constant negative geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connectio ...

s are similarly invariant: that is, geodesics map to geodesics under coordinate transformation.
Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane.
Once we choose a coordinate chart (one of the "models"), we can always embed
Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
* Embedding, in mathematics, one instance of some mathematical object contained within another instance
** Graph embedding
* Embedded generation, a distributed g ...

it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts; but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.
Since the four models describe the same metric space, each can be transformed into the other.
See, for example:
* the Beltrami–Klein model's relation to the hyperboloid model,
* the Beltrami–Klein model's relation to the Poincaré disk model,
* and the Poincaré disk model's relation to the hyperboloid model.
Isometries of the hyperbolic plane

Everyisometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' m ...

( transformation or motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer a ...

) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In ''n''-dimensional hyperbolic space, up to ''n''+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)
All the isometries of the hyperbolic plane can be classified into these classes:
* Orientation preserving
** the identity isometry — nothing moves; zero reflections; zero degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

.
** inversion through a point (half turn) — two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point; two degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

.
** rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

around a normal point — two reflections through lines passing through the given point (includes inversion as a special case); points move on circles around the center; three degrees of freedom.
** "rotation" around an Hyperbolic geometry in art

M. C. Escher's famous prints '' Circle Limit III'' and ''Circle Limit IV'' illustrate the conformal disc model ( Poincaré disk model) quite well. The white lines in ''III'' are not quite geodesics (they are hypercycles), but are close to them. It is also possible to see quite plainly the negativecurvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...

of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.
For example, in ''Circle Limit III'' every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...

. In ''Circle Limit III'', for example, one can see that the number of fishes within a distance of ''n'' from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius ''n'' must rise exponentially in ''n''.
The art of crochet
Crochet (; ) is a process of creating textiles by using a crochet hook to interlock loops of yarn, thread, or strands of other materials. The name is derived from the French term ''crochet'', meaning 'hook'. Hooks can be made from a variety o ...

has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, whose book '' Crocheting Adventures with Hyperbolic Planes'' won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.
HyperRogue
''HyperRogue'' is an independent video game developed by Zeno Rogue. It is a roguelike inspired by the puzzle game '' Deadly Rooms of Death'' and the art of M. C. Escher, taking place in the hyperbolic plane.
Gameplay
''HyperRogue'' is a ...

is a roguelike
Roguelike (or rogue-like) is a subgenre of role-playing computer games traditionally characterized by a dungeon crawl through procedurally generated levels, turn-based gameplay, grid-based movement, and permanent death of the player chara ...

game set on various tilings of the hyperbolic plane.
Higher dimensions

Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions.Homogeneous structure

Hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. T ...

of dimension ''n'' is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

to the quotient
:: $\backslash mathrm(1,n)/(\backslash mathrm(1)\; \backslash times\; \backslash mathrm(n)).$
The orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...

by norm-preserving transformations on Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the in ...

Rproduct
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Pr ...

of the orthogonal groups O(''n'') and O(1), where O(''n'') acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...

ic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.
In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms allow one to interpret the upper half plane model as the quotient and the Poincaré disc model as the quotient . In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism . This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipot ...

upper triangular matrices.
See also

* Band model * Constructions in hyperbolic geometry * Hjelmslev transformation * Hyperbolic 3-manifold * Hyperbolic manifold * Hyperbolic set * Hyperbolic tree * Kleinian group * Lambert quadrilateral * Open universe * Poincaré metric * Saccheri quadrilateral * Systolic geometry * Uniform tilings in hyperbolic plane * δ-hyperbolic spaceNotes

References

* A'Campo, Norbert and Papadopoulos, Athanase, (2012) ''Notes on hyperbolic geometry'', in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, SBN , DOI 10.4171–105. * Coxeter, H. S. M., (1942) ''Non-Euclidean geometry'', University of Toronto Press, Toronto * * * Lobachevsky, Nikolai I., (2010) ''Pangeometry'', Edited and translated by Athanase Papadopoulos, Heritage of European Mathematics, Vol. 4. Zürich: European Mathematical Society (EMS). xii, 310~p, /hbk * Milnor, John W., (1982)Hyperbolic geometry: The first 150 years

', Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24. * Reynolds, William F., (1993) ''Hyperbolic Geometry on a Hyperboloid'',

American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an ...

100:442–455.
*
* Samuels, David, (March 2006) ''Knit Theory'' Discover Magazine, volume 27, Number 3.
* James W. Anderson, ''Hyperbolic Geometry'', Springer 2005,
* James W. Cannon, William J. Floyd, Richard Kenyon, and Walter R. Parry (1997) Hyperbolic Geometry

', MSRI Publications, volume 31.

External links

University of New Mexico * ttps://www.youtube.com/watch?v=B16YjC9OS0k&mode=user&search= "The Hyperbolic Geometry Song"A short music video about the basics of Hyperbolic Geometry available at YouTube. * * *

More on hyperbolic geometry, including movies and equations for conversion between the different models

University of Illinois at Urbana-Champaign

Hyperbolic Voronoi diagrams made easy, Frank Nielsen

* , interactive instructional website.

Hyperbolic Planar Tesselations

Models of the Hyperbolic Plane

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