Hyperbolic geometry

Lines through a given point P and asymptotic to line R
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultra-parallel lines

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. In hyperbolic geometry the parallel postulate of Euclidean geometry 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 this with Playfair's axiom the modern version of Euclid's parallel postulate)

Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.

A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.

When geometers first realised they worked with something else than the standard Euclidean geometry they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry. It was for putting it in the now rarely used sequence elliptic geometry (spherical geometry) , parabolic geometry (Euclidean geometry), and hyperbolic geometry. In Russia it is commonly called Lobachevskian geometry after one of its discoverers, the Russian geometer Nikolai Lobachevsky.

This page is mainly about the 2 dimensional or plane hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry.

Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.


    Relation to Euclidean geometry

    Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. 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 hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.


    Lines through a given point P and asymptotic to line R.

    Hyperbolic geometry also has several properties that differ from those of Euclidean geometry. There are at least two distinct lines through a point P not on a line R that do not intersect R, where all lie in one plane. This means 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 them (x and y in the diagram) are limiting parallels: they asymptotically approach R. The remainder are called ultraparallel and will eventually diverge from R on both sides of P.

    The limiting parallels make an angle θ with PB; this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism.

    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.

    In hyperbolic geometry, there is no line that remains equidistant from another. Instead, the points that have all the same orthogonal distance from a given line are on a curve called a hypercycle.

    Circles and disks

    In hyperbolic geometry, the circumference of a circle of radius r is greater than .

    Let , where is the (negative) Gaussian curvature of the plane.

    Then the circumference of a circle of radius r is equal to:

    And the area of the enclosed disk is:

    Hypercycles and horocycles

    In hyperbolic geometry, there is no line that remains equidistant from another. Instead, the points that have all the same orthogonal distance from a given line are on a curve called a hypercycle.

    Another special curve is the horocycle, a curve whose normal radii are limiting parallel to each other (all converge asymptotically in one direction to the same ideal point). Through every pair of points there are 2 horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them.

    3 distinct points lie on either a line, a hypercycle, a horocycle, or a circle.

    The length of the line-segment is the shortest length between two points. The arc-length of an hypercycle connecting two points is longer that 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 the arc-length of a horocycles connecting two points.


    Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians. The difference is referred to as the defect.

    The area of a hyperbolic triangle is given by its defect in radians multiplied by R2. As a consequence, all hyperbolic triangles have an area that is less than R2π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.

    As in Euclidean geometry, each hyperbolic triangle has an inscribed circle. In hyperbolic geometry, if its vertices lie on an horocycle or hypercycle, a triangle has no circumscribed circle.

    As in spherical and elliptical geometry, in hyperbolic geometry the only similar triangles are congruent triangles.

    Standardized Gaussian curvature

    Though hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, 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 radians.
    • The length of an arc of an horocycle such that the tangent at one extremity is limiting parallel to the radius through the other extremity is 1.
    • The ratio of the arc lengths between two radii of two horocycles where the horocycles are a distance 1 apart is e : 1.

    Cartesian-like coordinate systems

    In hyperbolic geometry rectangles do not exist. (the sum of the angles of a rectangle is always less than 4 right angles) also there are no equidistant lines. (see hypercycles above). This all has influences on the 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 horocycle trough the origin centered around 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.


    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.


    Since the publication of Euclid's Elements circa 300 BCE, many geometers made attempts to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám,Nasīr al-Dīn al-Tūsī, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, 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 quadrilaterals, 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 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 János Bolyai, Nikolai Ivanovich Lobachevsky and Carl Friedrich Gauss. Other than their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, Gauss, Bolyai and Lobachevsky 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" causing several modern authors still consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Lobachevsky published in 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 Euclidean geometry was.

    The term "hyperbolic geometry" was introduced by Felix Klein in 1871. Klein followed an initiative of Arthur Cayley to use the transformations of projective geometry to produce isometries. The idea used a conic section or quadric to define a region, and used cross ratio to define a metric . The projective transformations that leave the conic section or quadric stable 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, and the references Coxeter and Milnor.

    Philosophical consequences

    The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers (for example Hobbes and Spinoza) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in Euclid's Elements.

    Kant in the Critique of Pure Reason came to the conclusion that space (in Euclidean geometry) 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 Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians", 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, analytical philosophy and logic. 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 the parallax of Sirius and treating Sirius as the ideal point of an angle of parallelism. 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 (7017299195741400000♠2000000 AU, 10 parsec). Some argue that his measurements were methodologically flawed.

    Henri Poincaré, with his sphere-world thought experiment, 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 places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately.Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity).

    In relativity, rather than considering Euclidean, hyperbolic and elliptic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space.

    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).

    Hyperbolic plane geometry as the geometry of saddle surfaces

    A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultra-parallel lines

    Hyperbolic plane geometry is the geometry of saddle surfaces with constant negative Gaussian curvature (for example the pseudosphere).

    By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space.

    There exist various pseudospherical surfaces that have a finite area of constant negative Gaussian curvature.

    Physical realizations of the hyperbolic plane

    The hyperbolic plane is a plane where every point is a saddle point. By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space.

    There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.

    Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the pseudosphere is due to William Thurston.

    A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring
    A coral with similar geometry on the Great Barrier Reef

    The art of crochet has been used (see Mathematics and fiber arts § Knitting and crochet) to demonstrate hyperbolic planes with the first being made by Daina Taimina.

    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, has been made available by Jeff Weeks.

    Models of the hyperbolic plane

    There are different that have for a large area a constant negative Gaussian curvature, the pseudosphere being the best well known of them.

    But it is easier to do hyperbolic geometry on other models.

    Poincaré disk model of truncated triheptagonal tiling
    Lines through a given point and parallel to a given line, illustrated in the Poincaré disk model

    There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. 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, Klain disk model and Klein model, is named after Eugenio Beltrami and Felix Klein.

    For the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane, and the chords of this circle are the hyperbolic lines.

    For higher dimensions this model uses the interior of the unit ball, and the chords of this n-ball are the hyperbolic lines.

    • This model has the advantage that lines are straight, but the disadvantage that angles are distorted (the mapping is not conformal), and also circles are not represented as circles.

    The Poincaré disk model

    The Poincaré disk model, also known as the conformal disk model, also employs the interior of the unit circle, but lines are represented by arcs of circles that are orthogonal 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 transformations.
    • Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than the hyperbolic center of the circle.
    • Horocycles are circles within the disk which are tangent 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

    The Poincaré half-plane model 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 and the x-axis 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.
    • Like the Poincaré disk model, this model preserves angles, and is thus conformal. All isometries within this model are therefore Möbius transformations 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

    The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds says that Wilhelm Killing used this model in 1885

    • This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time.
    • The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.
    • The model generalizes directly to an additional dimension, where three-dimensional hyperbolic geometry relates to Minkowski 4-space.

    Other models of the hyperbolic plane

    The hemisphere model

    The hemisphere 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:

    The hemisphere model is part of a Riemann sphere, and different projections give different models of the hyperbolic plane:

    • Central projection from the centre of the sphere onto the plane projects corresponding points on the

    See further: Connection between the models (below)

    The Gans Model

    The or In 1966 David Gans proposed this model in the journal American Mathematical Monthly It is an orthographic projection 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.
    • The lines in this model are interpreted as the branches of a hyperbola.

    Connection between the models

    Poincaré disk, hemispherical and hyperboloid models are related by stereographic projection from −1. Beltrami–Klein model is orthographic projection from hemispherical model. Poincaré half-plane model here projected from the hemispherical model by rays from left end of Poincaré disk model.

    All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic space. The characteristic feature of the hyperbolic space itself is that it has a constant negative Gaussian curvature, which is indifferent to the coordinate chart used. The geodesics 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 space.

    Once we choose a coordinate chart (one of the "models"), we can always embed 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.

    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 quite close to them. It is also possible to see quite plainly the negative curvature 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. 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 equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

    A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring

    The art of crochet has been used to demonstrate hyperbolic planes with the first being made by Daina Taimina, whose book Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.

    Higher dimensions

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

    Homogeneous structure

    Hyperbolic space of dimension n is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic to the quotient

    The orthogonal group O(1, n) acts by norm-preserving transformations on Minkowski space R1,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 product 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 algebraic 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 SO+(1, 2) ≅ PSL(2, R) ≅ PSU(1, 1) allow one to interpret the upper half plane model as the quotient SL(2, R)/SO(2) and the Poincaré disc model as the quotient SU(1, 1)/U(1). In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in PGL(2, C) 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 PGL(2, C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O+(1, 3) ≅ PGL(2, C). 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 upper triangular matrices.

    See also