Relation to Euclidean geometry

Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

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

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 the model used, even if the lines may look radically different.

Non-intersecting / parallel lines

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

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:

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

Circles and disks

In hyperbolic geometry, the circumference of a circle of radius r is greater than ${\displaystyle 2\pi r}$.

Let ${\displaystyle R={\frac {1}{\sqrt {-K}}}}$, where ${\displaystyle K}$ is the Gaussian curvature of the plane. In hyperbolic geometry, ${\displaystyle K}$ is negative, so the square root is of a positive number.

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

${\displaystyle 2\pi R\sinh {\frac {r}{R}}\,.}$

And the area of the enclosed disk is:

${\displaystyle 4\pi R^{2}\sinh ^{2}{\frac {r}{2R}}=2\pi R^{2}\left(\cosh {\frac {r}{R}}-1\right)\,.}$

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than ${\displaystyle 2\pi }$, 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: ${\displaystyle {\frac {1}{\tanh(r)}}}$^[1]

Hypercycles and horocycles

Hypercycle and pseudogon in the Poincare disk model

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 the horocycle, a curve whose normal radii (perpendicular lines) are all limiting parallel to each other (all converge asymptotically in one direction to the same ideal point, the centre of the horocycle).

Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them.

Given any three distinct points, they all lie on either a line, hypercycle, horocycle, 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.^[1]

Triangles

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 (180°, a straight angle). 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 or equal to 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 incircle. In hyperbolic geometry, if all three of its vertices lie on a horocycle or hypercycle, then the triangle has no circumscribed circle.

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

Regular apeirogon and pseudogon

An apeirogon and circumscribed horocycle in the Poincare disk model

Special polygons in hyperbolic geometry are the regular apeirogon and pseudogon uniform polygons with an infinite number of sides.

In Euclidean geometry, 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 or pseudogon has sides of any length (i.e., it remains a polygon with noticeable sides).

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 then it is an apeirogon and can be inscribed and circumscribed by concentric horocycles.

If the bisectors are diverging parallel then it is a pseudogon and can be inscribed and circumscribed by 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.)

Tessellations

Rhombitriheptagonal tiling of the hyperbolic plane, seen in the Poincaré disk model

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons 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.^[2]

Cartesian-like coordinate systems

Compared to Euclidean geometry, hyperbolic geometry presents numerous difficulties for a coordinate system: the sum of the angles of a quadrilateral is always less than 360 degrees; there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes it to rotate when it returns to the origin; etc.

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 through the origin centered around ${\displaystyle (0,+\infty )}$ and the length along this horocycle.^[4]

Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

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 signed distance 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

$${\displaystyle \operatorname {dist} (\langle x_{1},y_{1}\rangle ,\langle x_{2},y_{2}\rangle )=\operatorname {arcosh} \left(\cosh y_{1}\cosh(x_{2}-x_{1})\cosh y_{2}-\sinh y_{1}\sinh y_{2}\right)\,.}$$

This formula can be derived from the formulas about hyperbolic triangles.

The corresponding metric tensor field is: ${\displaystyle (\mathrm {d} s)^{2}=\cosh ^{2}y\,(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}$.

In this coordinate system, straight lines take one of these forms ((x, y) is a point on the line; x₀, y₀, A, and α are parameters):

ultraparallel to the x-axis

${\displaystyle \tanh(y)=\tanh(y_{0})\cosh(x-x_{0})}$

asymptotically parallel on the negative side

${\displaystyle \tanh(y)=A\exp(x)}$

asymptotically parallel on the positive side

${\displaystyle \tanh(y)=A\exp(-x)}$

intersecting perpendicularly

${\displaystyle x=x_{0}}$

intersecting at an angle α

${\displaystyle \tanh(y)=\tan(\alpha )\sinh(x-x_{0})}$

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.

19th-century developments

In the 19th century, hyperbolic geometry was explored extensively by Nikolai Ivanovich Lobachevsky, János Bolyai, Carl Friedrich Gauss 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.^[10][11] 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"^[12] 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 Euclidean geometry was.

The term "hyperbolic geometry" was introduced by Felix Klein in 1871.^[13] 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..."^[14]

For more history, see article on non-Euclidean geometry, and the references Coxeter^[15] and Milnor.^[16]

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.^[17]

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").^[18] 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 (2000000 AU, 10 parsec).^[19] Some argue that his measurements were methodologically flawed.^[20]

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.^[21]

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.^[22][23] Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity).^[24]

In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space,^[25][26] corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity through rapidity, which stands in for velocity, and is expressed by a hyperbolic angle. 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).^[27]

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 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 distance in this model is half the logarithm of the cross-ratio, which was introduced by Arthur Cayley in projective geometry.

The Poincaré disk model

Poincaré disk model with truncated triheptagonal tiling

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 is 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.
• The length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation ${\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\quad \lambda >0.}$
• 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^[33] 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 (positions in spacetime) that various inertially moving observers, starting from a common event, 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: a hyperbolic 3-space three-dimensional hyperbolic geometry relates to Minkowski 4-space.

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: ${\displaystyle x^{2}+y^{2}+z^{2}=1,z>0.}$

The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere.

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

• Stereographic projection from ${\displaystyle (0,0,-1)}$ onto the plane ${\displaystyle z=0}$ projects corresponding points on the Poincaré disk model
• Stereographic projection from ${\displaystyle (0,0,-1)}$ onto the surface ${\displaystyle x^{2}+y^{2}-z^{2}=-1,z>0}$ projects corresponding points on the hyperboloid model
• Stereographic projection from ${\displaystyle (-1,0,0)}$ onto the plane ${\displaystyle x=1}$ projects corresponding points on the Poincaré half-plane model
• Orthographic projection onto a plane ${\displaystyle z=C}$ projects corresponding points on the Beltrami–Klein model.
• Central projection from the centre of the sphere onto the plane ${\displaystyle z=1}$ 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.^[34] 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 represented as branches of a hyperbola.^[35]

The band model

The band model employs a portion of the Euclidean plane between two parallel lines.^[36] Distance is preserved along one line through the middle of the band. Assuming the band is given by ${\displaystyle \{z\in \mathbb {C}$ :|\operatorname {Im} z|<\pi /2\}} , the metric is given by ${\displaystyle |dz|\sec(\operatorname {Im} z)}$.

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 plane. The characteristic feature of the hyperbolic plane 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 plane.^[37]

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.