Editing Hyperbolic triangle

{{Short description|Triangle in hyperbolic geometry}}
{{About|[[triangle]]s in [[hyperbolic geometry]]|triangles in a hyperbolic sector|Hyperbolic sector#Hyperbolic triangle}}

[[File:Hyperbolic triangle.svg|thumb|250px|right|A hyperbolic triangle embedded in a [[saddle point|saddle-shaped surface]]]]

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.
[[File:Order-7 triangular tiling.svg|thumb|right|200px|An [[order-7 triangular tiling]] has equilateral triangles with 2&pi;/7 radian [[internal angle]]s.]]

==Definition==
A hyperbolic triangle consists of three non-[[collinear]] points and the 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>
<!-- ref>[[Svetlana Katok]] (1992) ''Fuchsian Groups'', [[University of Chicago Press]] {{ISBN|0-226-42583-5}}</ref> What namely says the book on the definition? -->

==Properties==
Hyperbolic triangles have some properties that are analogous to those of [[triangle]]s in [[Euclidean geometry]]:

*Each hyperbolic triangle has an [[inscribed circle]] but not every hyperbolic triangle has a [[circumscribed circle]] (see below). Its vertices can lie on a [[horocycle]] or [[hypercycle (geometry)|hypercycle]].

<!-- copied from [[spherical geometry       ]] and changed where needed -->
Hyperbolic triangles have some properties that are analogous to those of triangles in [[spherical geometry|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 [[inscribed circle]].
*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]] or when all of its vertices lie on a [[horocycle]] or on a one sided [[hypercycle (geometry)|hypercycle]].
*[[δ-hyperbolic space|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==

[[File:Ideal circles.svg|thumb|right|200px|Three ideal triangles in the [[Poincaré disk model]]]]

The definition of a triangle can be generalized, permitting vertices on the [[ideal point|ideal boundary]] of the plane while keeping the sides within the plane. If a pair of sides is ''[[limiting parallel]]'' (i.e. the distance between them approaches zero as they tend to the [[ideal point]], but they do not intersect), then they end at an '''ideal vertex''' represented as an ''[[ideal point|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 [[line (geometry)|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 angle|right]], one of the first hyperbolic triangles (1818) described by [[Ferdinand Karl Schweikart]].

===Ideal triangle===
{{Main|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 [[hyperbolic geometry#Standardized Gaussian curvature|absolute length]] (a special unit of length analogous to a relations between distances in [[spherical geometry]]). This choice for this length scale makes formulas simpler.<ref>{{cite book|last=Needham|first=Tristan|title=Visual Complex Analysis|publisher=Oxford University Press|year=1998|isbn=9780198534464|page=270|url=https://books.google.com/books?id=ogz5FjmiqlQC&pg=PA270}}</ref>

In terms of the [[Poincaré half-plane model]] absolute length corresponds to the [[Riemannian manifold|infinitesimal metric]] <math>ds=\frac{|dz|}{\operatorname{Im}(z)}</math> and in the [[Poincaré disk model]] to <math>ds=\frac{2|dz|}{1-|z|^2}</math>.

In terms of the (constant and negative) [[Gaussian curvature]] {{mvar|K}} of a hyperbolic plane, a unit of absolute length corresponds to a length of
:<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]]. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is 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&nbsp;{{mvar|R}}:
:<math>(\pi-A-B-C) R^2{}{}\!</math>.

This theorem, first proven by [[Johann Heinrich Lambert]],<ref>{{cite book|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=https://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> is related to [[Girard's theorem]] in spherical geometry.

==Trigonometry==
In all the formulas stated below the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} must be measured in [[Hyperbolic geometry#Standardized Gaussian curvature|absolute length]], a unit so that the [[Gaussian curvature]] {{mvar|K}} of the plane is −1. In other words, the quantity {{mvar|R}} in the paragraph above is supposed to be equal to 1.

Trigonometric formulas for hyperbolic triangles depend on the [[hyperbolic function]]s sinh, cosh, and tanh.

===Trigonometry of right triangles===
If ''C'' is a [[right angle]] then:
*The '''sine''' of angle ''A'' is the '''hyperbolic sine''' of the side opposite the angle divided by 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 '''hyperbolic tangent''' of the adjacent leg divided by 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 '''hyperbolic tangent''' of the opposite leg divided by the '''hyperbolic sine''' of the adjacent leg.
::<math>\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}} = \frac{\tanh a}{\,\sinh b\,}</math>.

*The '''hyperbolic cosine''' of the adjacent leg to angle A is the '''cosine''' of angle B divided by the '''sine''' of angle A.
::<math>\textrm{cosh(adjacent)}= \frac{\cos B}{\sin A}</math>.

*The '''hyperbolic cosine''' of the hypotenuse is the product of the '''hyperbolic cosines ''' of the legs.
::<math>\textrm{cosh(hypotenuse)}= \textrm{cosh(adjacent)} \textrm{cosh(opposite)}</math>.

*The '''hyperbolic cosine''' of the hypotenuse is also the product of the '''cosines''' of the angles divided by the product of their '''sines'''.<ref>{{cite book|last1=Martin|first1=George E.|title=The foundations of geometry and the non-Euclidean plane|url=https://archive.org/details/foundationsofgeo0000mart|url-access=registration|date=1998|publisher=Springer|location=New York, NY|isbn=0-387-90694-0|page=[https://archive.org/details/foundationsofgeo0000mart/page/433 433]|edition=Corrected 4. print.}}</ref>
::<math>\textrm{cosh(hypotenuse)}= \frac{\cos A \cos B}{\sin A\sin B} = \cot A \cot B</math>

====Relations between angles====
We also have the following equations:<ref>{{cite book|last1=Smogorzhevski|first1=A.S.|title=Lobachevskian geometry|publisher=Mir Publishers|location=Moscow 1982|page=63}}</ref>

:<math> \cos A = \cosh a \sin B</math>

:<math> \sin A = \frac{\cos B}{\cosh b}</math>

:<math> \tan A = \frac{\cot B}{\cosh c}</math>

:<math> \cos B = \cosh b \sin A</math>

:<math> \cosh c = \cot A \cot B</math>

====Area====
The area of a right angled triangle is:
:<math>\textrm{Area} = \frac{\pi}{2} - \angle A - \angle B</math>
also
:<math>\textrm{Area}= 2 \arctan (\tanh (\frac{a}{2})\tanh (\frac{b}{2}) )</math>{{citation needed|date=October 2015}}<ref>{{cite web|title=Area of a right angled hyperbolic triangle as function of side lengths|url=https://math.stackexchange.com/q/1462778 |website=[[Stack Exchange]] Mathematics|accessdate=11 October 2015}}</ref>
The area for any other triangle is:
:<math>\textrm{Area} = {\pi} - \angle A - \angle B - \angle C</math>

====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 = <math> \infty </math> and <math>\textrm{tanh}(\infty )= 1</math>, resulting in <math>\cos A= \textrm{tanh(adjacent)}</math>.

====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:

:<math>\cos A= \frac{\textrm{tanh}(\frac12 s) }{\textrm{tanh} (s)}</math>

:<math>\cosh( \frac12 s)= \frac{\cos(\frac12 A)}{\sin( A)}= \frac{1}{2 \sin(\frac12 A)}</math>

===General trigonometry===
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 [[duality (projective geometry)|dual theorem]] 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>
which is derived in the same way as the [[Spherical_trigonometry#Cotangent_four-part_formulae|analogous formula in spherical trigonometry]].

<!--- still in development
====Solving Hyperbolic triangles====
see also [[Solving triangles]]
*'''''Angle - Angle - Angle''''' use the dual form of the hyperbolic law of cosines
*'''''Angle - Angle - Side''''' use hyperbolic law of sines to get to Angle - Angle - Side -side
*'''''Angle - Angle - Side -side ''''' use the four-parts formula
*'''''Angle - Side - Angle'''''
*'''''Angle - Side - side''''' use hyperbolic law of sines to get to Angle - Angle - Side -side
*'''''Side - Angle - Side'''''
*'''''Side - Side - Side''''' use the hyperbolic law of cosines
end of still in development ---->

==See also==
*[[Pair of pants (mathematics)]]
*[[Triangle group]]
For hyperbolic trigonometry:
*[[Angle of parallelism]]
*[[Hyperbolic law of cosines]]
*[[Law of sines#Hyperbolic case|Hyperbolic law of sines]]
*[[Lambert quadrilateral]]
*[[Saccheri quadrilateral]]

==References==
{{Reflist}}

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

[[Category:Hyperbolic geometry|Triangle]]
[[Category:Types of triangles]]