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Task 2: Fix CS1 deprecated coauthor parameter errors

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:''Not to be confused with the'' [[hyperbolic sector#Hyperbolic triangle|hyperbolic triangle of a hyperbolic sector]]
In [[hyperbolic geometry]], a '''hyperbolic triangle''' is a [[triangle]] in the [[hyperbolic plane]]. It consists of three [[line segment]]s called ''sides'' or ''edges'' and three [[point (geometry)|points]] called ''angles'' or ''vertices''.

Just as in the [[Euclidean space|Euclidean]] case, three points of a [[hyperbolic space]] of an arbitrary [[dimension (mathematics)|dimension]] always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces.
[[Image:Uniform tiling 73-t2.png|thumb|right|200px|A tiling of the hyperbolic plane with hyperbolic triangles – the [[order-7 triangular tiling]].]]

==Definition==
A hyperbolic triangle consists of three non-[[collinear]] points and three segments between them.<ref>{{citation|first=Wilson|last=Stothers|title=Hyperbolic geometry|url=http://www.maths.gla.ac.uk/~wws/cabripages/hyperbolic/hyperbolic0.html|publisher=[[University of Glasgow]]|year=2000}}, interactive instructional website</ref> The relations among the angles and sides are analogous to those of [[spherical trigonometry]]; they are most conveniently stated if the lengths are measured in terms of a special unit of length analogous to a [[radian]].<ref>For instance, the absolute length scales for both spherical geometry (the radian) and hyperbolic geometry can be defined as the perimeters of equilateral triangles with fixed angular defects; see {{citation|title=Visual Complex Analysis|first=Tristan|last=Needham|publisher=Oxford University Press|year=1998|isbn=9780198534464|page=270|url=http://books.google.com/books?id=ogz5FjmiqlQC&pg=PA270}}. As with the radian, the choice of units for this length scale is the one that makes the area formula as simple as possible.</ref> In terms of the [[Gaussian curvature]] {{mvar|K}} of the plane this unit is given by

::<math>R=\frac{1}{\sqrt{-K}}.</math>

In a hyperbolic triangle the [[sum of angles of a triangle|sum of the angles]] ''A'', ''B'', ''C'' (respectively opposite to the side with the corresponding letter) is strictly less than a [[straight angle]]. This is contrasted to [[Euclidean triangle]]s where this sum is always equal to the straight angle, as well as to [[spherical triangle]]s where this sum is greater. The difference is often called the [[angular defect|defect]] of the triangle. The [[area]] of a hyperbolic triangle is equal to its defect multiplied by the [[square (algebra)|square]] of {{mvar|R}}:

::<math>(\pi-A-B-C) R^2{}{}.\!</math>

This theorem, first proven by [[Johann Heinrich Lambert]],<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=Graduate Texts in Mathematics|first=John|last=Ratcliffe|publisher=Springer|year=2006|isbn=9780387331973|page=99|url=http://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99|quotation=That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph ''Theorie der Parallellinien'', which was published posthumously in 1786.}}</ref> corresponds to [[Girard's theorem]] in spherical geometry. In all the formulas stated below the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} must be measured in this unit. In other words, {{mvar|R}} is supposed to be equal to 1.

==Ideal vertices==
[[File:H2checkers_iii.png|thumb|Ideal triangles in [[infinite-order triangular tiling]] [[Schläfli symbol|{3,∞}]] ]]
The definition of a triangle can be generalized, permitting for vertices outside the plane itself, but keeping sides within the plane. If a pair of sides is ''[[asymptote|asymptotic]]'' (i.e. distance between them [[vanish (mathematics)|vanishes]] but they do not intersect), then they end at an '''ideal vertex''' represented as an ''[[omega point (geometry)|omega point]]''. Such pair of sides may also be said to form an angle of [[zero]]. It is impossible in [[Euclidean geometry]] for [[line (geometry)|straight]] sides lying on distinct lines. Though, such zero angles are common with [[tangent circles]].

A triangle with one ideal vertex is called an '''omega triangle'''. If all three vertices are ideal, then the resulting figure is called an ''[[ideal triangle]]''. The latter is characterized by zero sum of the angles.

==Right triangles==
Trigonometry formulas for hyperbolic triangles depend on the [[hyperbolic functions]] sinh, cosh, and tanh:
If ''C'' is a right angle then:

*The '''sine''' of angle A is the ratio of the '''hyperbolic sine''' of the side opposite the angle to the '''hyperbolic sine''' of the [[hypotenuse]].
:: <math>\sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a}{\,\sinh c\,}.\,</math>
*The '''cosine''' of angle A is the ratio of the '''hyperbolic tangent''' of the adjacent leg to the '''hyperbolic tangent''' of the hypotenuse.
:: <math>\cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b}{\,\tanh c\,}.\,</math>
*The '''tangent''' of angle A is the ratio of the '''hyperbolic tangent''' of the opposite leg to the '''hyperbolic sine''' of the adjacent leg.
:: <math>\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}}=\frac{\tanh a}{\,\sinh b\,}.</math>

The instance of an ideal right triangle provides the configuration to examine the [[angle of parallelism]] in the triangle.

==Oblique triangles==
Whether ''C'' is a right angle or not, the following relationships hold:
The [[hyperbolic law of cosines]] is as follows:

:: <math>\cosh c=\cosh a\cosh b-\sinh a\sinh b \cos C,</math>

Its dual is

:: <math>\cos C= -\cos A\cos B+\sin A\sin B \cosh c,</math>

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

:<math>\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c},</math>

and a four-parts formula:

:: <math>\cos C\cosh a=\sinh a\coth b-\sin C\cot B.</math>

==See also==
* [[Pair of pants (mathematics)]]
* [[Triangle group]]

==References==
{{reflist}}

==Further reading==
* [[Svetlana Katok]] (1992) ''Fuchsian Groups'', [[University of Chicago Press]] ISBN 0-226-42583-5

{{DEFAULTSORT:Hyperbolic Triangle}}
[[Category:Hyperbolic geometry|Triangle]]
[[Category:Triangles]]

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en>Monkbot: Task 2: Fix CS1 deprecated coauthor parameter errors

en>MenoBot II: Bot: Auto-tagging and/or general fixes, removed stub tag