In hyperbolic geometry, a hyperbolic triangle is a triangle in the

hyperbolic plane 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'' ...

. It consists of three

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

s called ''sides'' or ''edges'' and three

points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

called ''angles'' or ''vertices''. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces. Order-7 triangular tiling

Definition

A hyperbolic triangle consists of three non- collinear points and the three segments between them.

Properties

Hyperbolic triangles have some properties that are analogous to those of triangles in Euclidean geometry: *Each hyperbolic triangle has an

inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

but not every hyperbolic triangle has a circumscribed circle (see below). Its vertices can lie on a

horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere ...

or hypercycle. Hyperbolic triangles have some properties that are analogous to those of triangles in spherical or elliptic geometry: *Two triangles with the same angle sum are equal in area. *There is an upper bound for the area of triangles. *There is an upper bound for radius of the

. *Two triangles are congruent if and only if they correspond under a finite product of line reflections. *Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent). Hyperbolic triangles have some properties that are the opposite of the properties of triangles in spherical or elliptic geometry: *The angle sum of a triangle is less than 180°. *The area of a triangle is proportional to the deficit of its angle sum from 180°. Hyperbolic triangles also have some properties that are not found in other geometries: *Some hyperbolic triangles have no circumscribed circle, this is the case when at least one of its vertices is an

ideal point In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' ...

or when all of its vertices lie on a

or on a one sided hypercycle. * Hyperbolic triangles are thin, there is a maximum distance δ from a point on an edge to one of the other two edges. This principle gave rise to δ-hyperbolic space.

Triangles with ideal vertices

The definition of a triangle can be generalized, permitting vertices on the ideal boundary of the plane while keeping the sides within the plane. If a pair of sides is ''

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

'' (i.e. the distance between them approaches zero as they tend to the

, but they do not intersect), then they end at an ideal vertex represented as an '' omega point''. Such a pair of sides may also be said to form an angle of zero. A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines. However, such zero angles are possible with tangent circles. A triangle with one ideal vertex is called an omega triangle. Special Triangles with ideal vertices are:

Triangle of parallelism

A triangle where one vertex is an ideal point, one angle is right: the third angle is the angle of parallelism for the length of the side between the right and the third angle.

Schweikart triangle

The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by

Ferdinand Karl Schweikart Ferdinand Karl Schweikart (1780–1857) was a German jurist and amateur mathematician who developed an ''astral geometry'' before the discovery of non-Euclidean geometry. Life and work Schweikart, son of an attorney in Hesse, was educated in th ...

Ideal triangle

The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles.

Standardized Gaussian curvature

The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles. The length scale is most convenient if the lengths are measured in terms of the absolute length (a special unit of length analogous to a relations between distances in

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

). This choice for this length scale makes formulas simpler. In terms of the

Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré ha ...

absolute length corresponds to the infinitesimal metric

ds=\frac

and in the

Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk th ...

ds=\frac

. In terms of the (constant and negative)

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...

of a hyperbolic plane, a unit of absolute length corresponds to a length of :

R=\frac

. In a hyperbolic triangle the sum of the angles ''A'', ''B'', ''C'' (respectively opposite to the side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of : :

(\pi-A-B-C) R^2\!

. This theorem, first proven by Johann Heinrich Lambert, is related to

Girard's theorem Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

in spherical geometry.

Trigonometry

In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the

of the plane is −1. In other words, the quantity in the paragraph above is supposed to be equal to 1. Trigonometric formulas for hyperbolic triangles depend on the hyperbolic functions sinh, cosh, and tanh.

Trigonometry of right triangles

If ''C'' is a

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

then: *The sine of angle ''A'' is the hyperbolic sine of the side opposite the angle divided by the hyperbolic sine of the hypotenuse. ::

\sin A=\frac=\frac.\,

*The cosine of angle ''A'' is the hyperbolic tangent of the adjacent leg divided by the hyperbolic tangent of the hypotenuse. ::

\cos A=\frac=\frac.\,

*The tangent of angle ''A'' is the hyperbolic tangent of the opposite leg divided by the hyperbolic sine of the adjacent leg. ::

\tan A=\frac = \frac

. *The hyperbolic cosine of the adjacent leg to angle A is the cosine of angle B divided by the sine of angle A. ::

\textrm= \frac

. *The hyperbolic cosine of the hypotenuse is the product of the hyperbolic cosines of the legs. ::

\textrm= \textrm \textrm

. *The hyperbolic cosine of the hypotenuse is also the product of the cosines of the angles divided by the product of their sines. ::

\textrm= \frac = \cot A \cot B

Relations between angles

We also have the following equations: :

\cos A = \cosh a \sin B

\sin A = \frac

\tan A = \frac

\cos B = \cosh b \sin A

\cosh c = \cot A \cot B

Area

The area of a right angled triangle is: :

\textrm = \frac - \angle A - \angle B

The area for any other triangle is: :

\textrm =  - \angle A - \angle B - \angle C

also :

\textrm= 2 \arctan (\tanh (\frac)\tanh (\frac) )

Angle of parallelism

The instance of an omega triangle with a right angle provides the configuration to examine the angle of parallelism in the triangle. In this case angle ''B'' = 0, a = c =

\infty

and

\textrm(\infty )= 1

, resulting in

\cos A= \textrm

Equilateral triangle

The trigonometry formulas of right triangles also give the relations between the sides ''s'' and the angles ''A'' of an equilateral triangle (a triangle where all sides have the same length and all angles are equal). The relations are: :

\cos A= \frac

\cosh( \frac12 s)= \frac= \frac

General trigonometry

Whether ''C'' is a right angle or not, the following relationships hold: The

hyperbolic law of cosines In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigono ...

is as follows: :

\cosh c=\cosh a\cosh b-\sinh a\sinh b \cos C,

Its dual theorem is :

\cos C= -\cos A\cos B+\sin A\sin B \cosh c,

There is also a ''law of sines'': :

\frac = \frac = \frac,

and a four-parts formula: :

\cos C\cosh a=\sinh a\coth b-\sin C\cot B

which is derived in the same way as the analogue formula in spherical trigonometry.