Hyperbolic geometry

From Wikipedia, the free encyclopedia

Lines through a given point P and asymptotic to line R

 

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

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or 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 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 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.

This page is mainly about the 2-dimensional (planar) 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.

Properties

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

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 lines can be infinitely extended.

Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two lines can intersect in no more than one point, intersecting lines have equal opposite angles, and adjacent angles of intersecting lines are supplementary.

When we add a third line then there are properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given 2 intersecting lines there are infinitely many lines that do not intersect either of the given lines.

These properties all are 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 parallel lines instead of limiting parallel lines, with ultraparallel lines being 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:

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

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: ${\displaystyle {\frac {1}{\tanh(r)}}}$

Hypercycles and horocycles

Hypercycle and pseudogon in the Poincare disk model

In hyperbolic geometry, there is no line that remains equidistant from another. 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 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 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

An apeirogon and circumscribed horocycle in the Poincare disk model

A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon 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 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 horocycles.

If the bisectors are diverging parallel the apeirogon (sometimes called an pseudogon) 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.

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 area of a horocyclic sector is equal to the length of its horocyclic arc.
• An arc of an 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.

Cartesian-like coordinate systems

In hyperbolic geometry the sum of the angles of a quadrilateral 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 horocycle through the origin centered around ${\displaystyle (0,+\infty )}$ 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.

Distance

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

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

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

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

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

${\displaystyle \tanh y=A\cosh x+B\sinh x\quad {\text{ when }}\quad A^{2}<1 annotation="">$

where A and B are real parameters which characterize the straight line.

History

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, 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..."

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 (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, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space, corresponding to zero, positive and negative curvature respectively.

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 a saddle point. There exist various pseudospheres 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 Gaussian curvature) in a three-dimensional Euclidean space.

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 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, have been made available by Jeff Weeks.

Models of the hyperbolic plane

There are different pseudospherical surfaces 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 with 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, 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

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 ${\displaystyle (x,y)\mapsto (x,\lambda y),\quad \lambda >0.}$
• 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 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.

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 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

The Gans model

The Gans model or flattened hyperboloid model: 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.

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 ${\displaystyle \{z\in \mathbb {C} :|\operatorname {Im} z|<\pi /2\}}$, 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.

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.

Isometries of the hyperbolic plane

Every isometry (transformation or motion) 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.
  • 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.
  • rotation 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 ideal point (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.^{[citation needed]}

Hyperbolic geometry in art

M.C. Escher's Circle Limit III, 1959

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 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.

The art of crochet has been used to demonstrate hyperbolic planes (pictured above) 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.

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

${\displaystyle \mathrm {O} (1,n)/(\mathrm {O} (1)\times \mathrm {O} (n)).}$

The orthogonal group O(1, n) acts by norm-preserving transformations on Minkowski space R^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 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.

History of trigonometry

From Wikipedia, the free encyclopedia

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE). During the Middle Ages, the study of trigonometry continued in Islamic mathematics, hence it was adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).

Etymology

The term "trigonometry" was derived from Greek τρίγωνον trigōnon, "triangle" and μέτρον metron, "measure".

The modern word "sine" is derived from the Latin word sinus, which means "bay", "bosom" or "fold" is indirectly, via Indian, Persian and Arabic transmission, derived from the Greek term khordḗ "bow-string, chord". The Greek term was adopted into Sanskrit as jyā "bow-string", later also in the variant jīvā. Sanskrit jīvā was rendered adopted into Arabic as jiba, written jb جب. This was then interpreted as the genuine Arabic word jayb, meaning "bosom, fold, bay", either by the Arabs or by a mistake of the European translators such as Robert of Chester, who translated jayb into Latin as sinus. Particularly Fibonacci's sinus rectus arcus proved influential in establishing the term sinus. The words "minute" and "second" are derived from the Latin phrases partes minutae primae and partes minutae secundae. These roughly translate to "first small parts" and "second small parts".

Development

Ancient Near East

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. However, as pre-Hellenic societies lacked the concept of an angle measure, they were limited to studying the sides of triangles instead.

The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere. Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants. There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.

The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.^[4] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.

Classical antiquity

The chord of an angle subtends the arc of the angle.

Ancient Greek and Hellenistic mathematicians made use of the chord. Given a circle and an arc on the circle, the chord is the line that subtends the arc. A chord's perpendicular bisector passes through the center of the circle and bisects the angle. One half of the bisected chord is the sine of one half the bisected angle, that is,

${\displaystyle \mathrm {chord} \ \theta =2\sin {\frac {\theta }{2}},}$

and consequently the sine function is also known as the half-chord. Due to this relationship, a number of trigonometric identities and theorems that are known today were also known to Hellenistic mathematicians, but in their equivalent chord form.

Although there is no trigonometry in the works of Euclid and Archimedes, in the strict sense of the word, there are theorems presented in a geometric way (rather than a trigonometric way) that are equivalent to specific trigonometric laws or formulas. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute angles, respectively. Theorems on the lengths of chords are applications of the law of sines. And Archimedes' theorem on broken chords is equivalent to formulas for sines of sums and differences of angles. To compensate for the lack of a table of chords, mathematicians of Aristarchus' time would sometimes use the statement that, in modern notation, sin α/sin β < α/β < tan α/tan β whenever 0° < β < α < 90°, now known as Aristarchus's inequality.

The first trigonometric table was apparently compiled by Hipparchus of Nicaea (180 – 125 BCE), who is now consequently known as "the father of trigonometry." Hipparchus was the first to tabulate the corresponding values of arc and chord for a series of angles.

Although it is not known when the systematic use of the 360° circle came into mathematics, it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 BC), since he measured an angle in terms of a fraction of a quadrant. It seems that the systematic use of the 360° circle is largely due to Hipparchus and his table of chords. Hipparchus may have taken the idea of this division from Hypsicles who had earlier divided the day into 360 parts, a division of the day that may have been suggested by Babylonian astronomy. In ancient astronomy, the zodiac had been divided into twelve "signs" or thirty-six "decans". A seasonal cycle of roughly 360 days could have corresponded to the signs and decans of the zodiac by dividing each sign into thirty parts and each decan into ten parts. It is due to the Babylonian sexagesimal numeral system that each degree is divided into sixty minutes and each minute is divided into sixty seconds.

Menelaus' theorem

Menelaus of Alexandria (ca. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles. He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles. Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°. Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus". He further gave his famous "rule of six quantities".

Later, Claudius Ptolemy (ca. 90 – ca. 168 AD) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The Almagest is primarily a work on astronomy, and astronomy relies on trigonometry. Ptolemy's table of chords gives the lengths of chords of a circle of diameter 120 as a function of the number of degrees n in the corresponding arc of the circle, for n ranging from 1/2 to 180 by increments of 1/2. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity. A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine. Ptolemy further derived the equivalent of the half-angle formula

${\displaystyle \sin ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos(x)}{2}}.}$

Ptolemy used these results to create his trigonometric tables, but whether these tables were derived from Hipparchus' work cannot be determined.

Neither the tables of Hipparchus nor those of Ptolemy have survived to the present day, although descriptions by other ancient authors leave little doubt that they once existed.

Indian mathematics

Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya. The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.

In the 7th century, Bhaskara the First produced a formula for calculating the sine of an acute angle without the use of a table. He also gave the following approximation formula for sin(x), which had a relative error of less than 1.9%:

${\displaystyle \sin x\approx {\frac {16x(\pi -x)}{5\pi ^{2}-4x(\pi -x)}},\qquad \left(0\leq x\leq \pi \right).}$

Later in the 7th century, Brahmagupta redeveloped the formula

${\displaystyle \ 1-\sin ^{2}(x)=\cos ^{2}(x)=\sin ^{2}\left({\frac {\pi }{2}}-x\right)}$

(also derived earlier, as mentioned above) and the Brahmagupta interpolation formula for computing sine values.

Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent. Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the θ, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.

No.	Series	Name	Western discoverers of the series and approximate dates of discovery
1	sin x  =  x − x3 / 3! + x5 / 5! − x7 / 7! + ...	Madhava's sine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
2	cos x  = 1 − x2 / 2! + x4 / 4! − x6 / 6! + ...	Madhava's cosine series	Isaac Newton (1670) and Wilhelm Leibniz (1676)
3	tan−1x  =  x − x3 / 3 + x5 / 5 − x7 / 7 + ...	Madhava's arctangent series	James Gregory (1671) and Wilhelm Leibniz (1676)

The Indian text the Yuktibhāṣā contains proof for the expansion of the sine and cosine functions and the derivation and proof of the power series for inverse tangent, discovered by Madhava. The Yuktibhāṣā also contains rules for finding the sines and the cosines of the sum and difference of two angles.

Chinese mathematics

Guo Shoujing (1231–1316)

In China, Aryabhata's table of sines were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the earlier Greek, Hellenistic, Indian and Islamic worlds. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. However, this embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations. The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc s of a circle given the diameter d, sagitta v, and length c of the chord subtending the arc, the length of which he approximated as

${\displaystyle s=c+{\frac {2v^{2}}{d}}.}$

Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).^[28] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610).^[31]

Middle Ages

Page from The Compendious Book on Calculation by Completion and Balancing by Muhammad ibn Mūsā al-Khwārizmī (c. AD 820)

Previous works were later translated and expanded in the medieval Islamic world by Muslim mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem. According to E. S. Kennedy, it was after this development in Islamic mathematics that "the first real trigonometry emerged, in the sense that only then did the object of study become the spherical or plane triangle, its sides and angles."

Methods dealing with spherical triangles were also known, particularly the method of Menelaus of Alexandria, who developed "Menelaus' theorem" to deal with spherical problems. However, E. S. Kennedy points out that while it was possible in pre-Islamic mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, astronomers initially used Menelaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method.

In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry. In 830 AD, Habash al-Hasib al-Marwazi produced the first table of cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.

By the 10th century AD, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. He also developed the following trigonometric formula:

${\displaystyle \ \sin(2x)=2\sin(x)\cos(x)}$

In his original text, Abū al-Wafā' states: "If we want that, we multiply the given sine by the cosine minutes, and the result is half the sine of the double". Abū al-Wafā also established the angle addition and difference identities presented with complete proofs:

${\displaystyle \sin(\alpha \pm \beta )={\sqrt {\sin ^{2}\alpha -(\sin \alpha \sin \beta )^{2}}}\pm {\sqrt {\sin ^{2}\beta -(\sin \alpha \sin \beta )^{2}}}}$

${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$

For the second one, the text states: "We multiply the sine of each of the two arcs by the cosine of the other minutes. If we want the sine of the sum, we add the products, if we want the sine of the difference, we take their difference".

He also discovered the law of sines for spherical trigonometry:

${\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.}$

Also in the late 10th and early 11th centuries AD, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:

${\displaystyle \cos a\cos b={\frac {\cos(a+b)+\cos(a-b)}{2}}}$

Al-Jayyani (989–1079) of al-Andalus wrote The book of unknown arcs of a sphere, which is considered "the first treatise on spherical trigonometry". It "contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.

The method of triangulation was first developed by Muslim mathematicians, who applied it to practical uses such as surveying and Islamic geography, as described by Abu Rayhan Biruni in the early 11th century. Biruni himself introduced triangulation techniques to measure the size of the Earth and the distances between various places. In the late 11th century, Omar Khayyám (1048–1131) solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. In the 13th century, Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.

In 1342, Levi ben Gershon, known as Gersonides, wrote On Sines, Chords and Arcs, in particular proving the sine law for plane triangles and giving five-figure sine tables.

A simplified trigonometric table, the "toleta de marteloio", was used by sailors in the Mediterranean Sea during the 14th-15th Centuries to calculate navigation courses. It is described by Ramon Llull of Majorca in 1295, and laid out in the 1436 atlas of Venetian captain Andrea Bianco.

In the 15th century, Jamshīd al-Kāshī provided the first explicit statement of the law of cosines in a form suitable for triangulation. In France, the law of cosines is still referred to as the theorem of Al-Kashi. He also gave trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg also gives accurate tables of sines and tangents correct to 8 decimal places around the same time.

Early modern mathematics

Regiomontanus was perhaps the first mathematician in Europe to treat trigonometry as a distinct mathematical discipline, in his De triangulis omnimodis written in 1464, as well as his later Tabulae directionum which included the tangent function, unnamed. The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In the 17th century, Isaac Newton and James Stirling developed the general Newton–Stirling interpolation formula for trigonometric functions.

In the 18th century, Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, deriving their infinite series and presenting "Euler's formula" e^ix = cos x + i sin x. Euler used the near-modern abbreviations sin., cos., tang., cot., sec., and cosec. Prior to this, Roger Cotes had computed the derivative of sine in his Harmonia Mensurarum (1722). Also in the 18th century, Brook Taylor defined the general Taylor series and gave the series expansions and approximations for all six trigonometric functions. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were also very influential in the development of trigonometric series.