Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai-Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

For any given line R and point P 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, the modern version of Euclid's parallel postulate.)

The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.

The hyperboloid model of hyperbolic geometry provides a representation of events one temporal unit into the future in Minkowski space, the basis of special relativity. Each of these events corresponds to a rapidity in some direction.

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 finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry. In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.

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.

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.

In hyperbolic geometry, the circumference of a circle of radius r is greater than \(2 \pi r\).

Let \(R = \frac{1}{\sqrt{-K}}\), where \(K\) is the Gaussian curvature 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:

\(2\pi R \sinh \frac{r}{R} \,.\)

And the area of the enclosed disk is:

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

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

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

Another special curve is the horocycle, 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 a 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 the arc horocycle, connecting the same two points.

The lengths of the arcs of both horocycles connecting two points are equal, and are longer than the arclength of any hypercycle connecting the points and shorter than the arc of any circle connecting the two points.

Condensed: the full section is in Wikipedia.

Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians (180°). The difference is called the defect. Generally, the defect of a convex hyperbolic polygon with \(n\) sides is its angle sum subtracted from \((n - 2) \cdot 180^\circ\).

The area of a hyperbolic triangle is given by its defect in radians multiplied by R, which is also true for all convex hyperbolic polygons. Therefore, all hyperbolic triangles have an area less than or equal to Rπ. 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 space, 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.

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° 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. 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 (since all its vertices are the same distance from a line, the axis, and the midpoints of its sides are also equidistant from that same axis).

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.

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 area of a horocyclic sector is equal to the length of its horocyclic arc.
• An arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1.
• The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e : 1.

Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper 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 Lobachevsky 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 \((0, + \infty )\) and the length along this horocycle.

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

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 \[\operatorname{dist} (\langle x_1, y_1 \rangle, \langle x_2, y_2 \rangle) = \operatorname{arcosh} \left( \cosh y_1 \cosh (x_2 - x_1) \cosh y_2 - \sinh y_1 \sinh y_2 \right) \,.\]

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

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

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

\(\tanh (y) = \tanh (y_0) \cosh (x - x_0)\)

asymptotically parallel on the negative side

\(\tanh (y) = A \exp (x)\)

\(\tanh (y) = A \exp (- x)\)

\(x = x_0\)

\(\tanh (y) = \tan (\alpha) \sinh (x - x_0)\)

Condensed: the full section is in Wikipedia.

Since the publication of Euclid's Elements around 300 BC, many geometers tried 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.

In the 19th century, hyperbolic geometry was explored extensively by Nikolai 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.

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 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 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..."

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

Kant in Critique of Pure Reason concluded 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" (stereotyped as dullards by the ancient Athenians), 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.