Inhyperbolic geometry, ahyperbolic triangle is atriangle in thehyperbolic plane. It consists of threeline segments calledsides oredges and threepoints calledangles orvertices.

Just as in theEuclidean case, three points of ahyperbolic space of an arbitrarydimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces.

A hyperbolic triangle consists of three non-collinear points and the three segments between them.^[1]

Hyperbolic triangles have some properties that are analogous to those oftriangles inEuclidean geometry:

• Each hyperbolic triangle has aninscribed circle but not every hyperbolic triangle has acircumscribed circle (see below). Its vertices can lie on ahorocycle orhypercycle.

Hyperbolic triangles have some properties that are analogous to those of triangles inspherical orelliptic 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 theinscribed circle.
• Two triangles are congruentif 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 nocircumscribed circle, this is the case when at least one of its vertices is anideal point or when all of its vertices lie on ahorocycle or on a one-sidedhypercycle.
• 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 toδ-hyperbolic space.

The definition of a triangle can be generalized, permitting vertices on theideal boundary of the plane while keeping the sides within the plane. If a pair of sides islimiting parallel (i.e. the distance between them approaches zero as they tend to theideal point, but they do not intersect), then they end at anideal vertex represented as anomega point.

Such a pair of sides may also be said to form an angle ofzero.

A triangle with a zero angle is impossible inEuclidean geometry forstraight sides lying on distinct lines. However, such zero angles are possible withtangent circles.

A triangle with one ideal vertex is called anomega triangle.

Special Triangles with ideal vertices are:

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

The triangle where two vertices are ideal points and the remaining angle isright, one of the first hyperbolic triangles (1818) described byFerdinand Karl Schweikart.

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

The relations among the angles and sides are analogous to those ofspherical trigonometry; thelength 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 theabsolute length (a special unit of length analogous to a relations between distances inspherical geometry). This choice for this length scale makes formulas simpler.^[2]

In terms of thePoincaré half-plane model absolute length corresponds to theinfinitesimal metric${\displaystyle ds=\frac{|dz|}{\mathrm{Im}(z)}}$ and in thePoincaré disk model to${\displaystyle ds=\frac{2|dz|}{1-|z{|}^2}}$.

In terms of the (constant and negative)Gaussian curvatureK of a hyperbolic plane, a unit of absolute length corresponds to a length of

${\displaystyle R=\frac{1}{\sqrt{-K}}}$.

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

${\displaystyle (\pi -A-B-C)R^2}$.

This theorem, first proven byJohann Heinrich Lambert,^[3] is related toGirard's theorem in spherical geometry.

A hyperbolic triangle is formed by three hyperbolic lines. These are the straight lines to an observer in the hyperbolic geometry and, given the metric, are the shortest distance paths between pairs of points on the lines. These lines are easily visualized using the half plane and thePoincaré disk planar models^[4][5] of the hyperbolic plane. For an observer in the hyperbolic plane, every position looks the same and there is no preferred direction. A figure can be moved from place to place without changing shape from the observer's viewpoint.

In the half plane model, points with positive imaginary part in thecomplex plane comprise the hyperbolic plane. The real axis is part of the boundary at infinity. Hyperbolic lines are the parts of circles or lines in the upper half complex plane with${\displaystyle \mathfrak{I}\mathfrak{m}z>0}$ that intersect the real axis at right angles.Möbius transformations transform extended circles to extended circles and preserve angles between intersecting circles. Extended circles are circles and extended lines. Extended lines include the point at infinity. Proper Möbius transformations not involving complex conjugation that transform hyperbolic lines into hyperbolic lines form the subgroup${\displaystyle {\text{Möb}}^{+}(\mathbb{H})}$ and have the form$${\displaystyle M(z)=\frac{az+b}{cz+d}}$$

witha,b,c,d real and with${\displaystyle ad-bc=1}$.${\displaystyle {\text{Möb}}^{+}(\mathbb{H})}$ preserves hyperbolic distances. The negative of thecomplex conjugate given by${\displaystyle M(z)=-\overline{z}}$ maps the real axis into itself and maps the upper half complex plane into itself and transforms hyperbolic lines into hyperbolic lines. When${\displaystyle {\text{Möb}}^{+}(\mathbb{H})}$ is extended with this transformation, the group${\displaystyle \text{Möb}(\mathbb{H})}$ results. Complex conjugation changes the signs of intersection angles of extended circles, but not their magnitudes.

In the Poincaré disk model, the boundary at infinity is theunit circle${\displaystyle |z|=1}$ in the complex plane and the points in the interior comprise the hyperbolic plane. Hyperbolic lines are the interior parts of extended circles intersecting the unit circle at right angles. The Möbius transformations transforming hyperbolic lines into hyperbolic lines form a group and preserve hyperbolic distance. The proper Mobius transformations in the subgroup${\displaystyle {\text{Möb}}^{+}(\mathbb{D})}$ have the form$${\displaystyle M(z)=\frac{\alpha z+\beta }{\overline{\beta }z+\overline{\alpha }}}$$with${\displaystyle |\alpha {|}^2-|\beta {|}^2=1}$. Here${\displaystyle \overline{\alpha }}$ denotes the complex conjugate of${\displaystyle \alpha }$ and, similarly,${\displaystyle \overline{\beta }}$ denotes the complex conjugate of${\displaystyle \beta }$. Complex conjugation${\displaystyle M(z)=\overline{z}}$ maps the unit circle into itself and maps the interior of the unit circle into itself and maps hyperbolic lines into hyperbolic lines. Angles of intersection change sign but their magnitudes are preserved. Combined with the proper subgroup${\displaystyle {\text{Möb}}^{+}(\mathbb{D})}$, these form the subgroup${\displaystyle \text{Möb}(\mathbb{D})}$.

There are Möbius transformations which take the interior of the unit circle in the complex plane into the upper complex plane${\displaystyle \mathfrak{I}\mathfrak{m}(z)>0}$. Their inverses do the reverse. Results in the Poincaré disk model can be carried over to the upper half plane planar model and vice versa. One such transformation is:

${\displaystyle \xi (z)=\frac{\frac{i}{\sqrt{2}}z+\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}z-\frac{i}{\sqrt{2}}}}$

The unit circle${\displaystyle |z|=1}$ is taken into the real axis${\displaystyle \mathfrak{I}\mathfrak{m}(z)=0}$ and the interior of the unit circle is taken into the upper half plane${\displaystyle \mathfrak{I}\mathfrak{m}(z)>0}$. This is easily seen. Three points determine an extended circle so since${\displaystyle \xi (-i)=\mathrm{\infty }\text{,}\xi (i)=0\text{,}\xi (1)=-1}$ and since a Mobius transformation maps extended circles into extended circles, it follow that the unit circle is taken into the extended real axis. An extended line includes the point at infinity. And since${\displaystyle \xi (0)=i}$, it follows by continuity that the interior of the unit circle is mapped into the upper half complex plane. The metric also can be shown to map correctly.

In all the formulas stated below the sidesa,b, andc must be measured inabsolute length, a unit so that theGaussian curvatureK of the plane is −1. In other words, the quantityR in the paragraph above is supposed to be equal to 1.

Trigonometric formulas for hyperbolic triangles depend on thehyperbolic functions sinh, cosh, and tanh.

IfC is aright angle then:

• Thesine of angleA is thehyperbolic sine of the side opposite the angle divided by thehyperbolic sine of thehypotenuse.

${\displaystyle \sin A=\frac{\text{sinh(opposite)}}{\text{sinh(hypotenuse)}}=\frac{\sinh a}{\sinh c}.}$

• Thecosine of angleA is thehyperbolic tangent of the adjacent leg divided by thehyperbolic tangent of the hypotenuse.

${\displaystyle \cos A=\frac{\text{tanh(adjacent)}}{\text{tanh(hypotenuse)}}=\frac{\tanh b}{\tanh c}.}$

• Thetangent of angleA is thehyperbolic tangent of the opposite leg divided by thehyperbolic sine of the adjacent leg.

${\displaystyle \tan A=\frac{\text{tanh(opposite)}}{\text{sinh(adjacent)}}=\frac{\tanh a}{\sinh b}}$.

• Thehyperbolic cosine of the adjacent leg to angle A is thecosine of angle B divided by thesine of angle A.

${\displaystyle \text{cosh(adjacent)}=\frac{\cos B}{\sin A}}$.

• Thehyperbolic cosine of the hypotenuse is the product of thehyperbolic cosines of the legs.

${\displaystyle \text{cosh(hypotenuse)}=\text{cosh(adjacent)}\text{cosh(opposite)}}$.

• Thehyperbolic cosine of the hypotenuse is also the product of thecosines of the angles divided by the product of theirsines.^[6]

${\displaystyle \text{cosh(hypotenuse)}=\frac{\cos A\cos B}{\sin A\sin B}=\cot A\cot B}$

We also have the following equations:^[7]

${\displaystyle \cos A=\cosh a\sin B}$

${\displaystyle \sin A=\frac{\cos B}{\cosh b}}$

${\displaystyle \tan A=\frac{\cot B}{\cosh c}}$

${\displaystyle \cos B=\cosh b\sin A}$

${\displaystyle \cosh c=\cot A\cot B}$

The area of a right angled triangle is:

${\displaystyle \text{Area}=\frac{\pi }{2}-\mathrm{\angle }A-\mathrm{\angle }B}$

also

${\displaystyle \text{Area}=2\arctan (\tanh (\frac{a}{2})\tanh (\frac{b}{2}))}$^{[citation needed][8]}

The area for any other triangle is:

${\displaystyle \text{Area}=\pi -\mathrm{\angle }A-\mathrm{\angle }B-\mathrm{\angle }C}$

The instance of anomega triangle with a right angle provides the configuration to examine theangle of parallelism in the triangle.

In this case angleB = 0, a = c =${\displaystyle \mathrm{\infty }}$ and${\displaystyle \text{tanh}(\mathrm{\infty })=1}$, resulting in${\displaystyle \cos A=\text{tanh(adjacent)}}$.

The trigonometry formulas of right triangles also give the relations between the sidess and the anglesA of anequilateral triangle (a triangle where all sides have the same length and all angles are equal).

The relations are:

${\displaystyle \cos A=\frac{\text{tanh}(\frac{1}{2}s)}{\text{tanh}(s)}}$

${\displaystyle \cosh (\frac{1}{2}s)=\frac{\cos (\frac{1}{2}A)}{\sin (A)}=\frac{1}{2\sin (\frac{1}{2}A)}}$

WhetherC is a right angle or not, the following relationships hold:Thehyperbolic law of cosines is as follows:

${\displaystyle \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos C,}$

Itsdual theorem is

${\displaystyle \cos C=-\cos A\cos B+\sin A\sin B\cosh c,}$

There is also alaw of sines:

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

and a four-parts formula:

${\displaystyle \cos C\cosh a=\sinh a\coth b-\sin C\cot B}$

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

It is easily observed that thetwo laws of cosines and the law of sines for spherical trigonometry are the same as those of hyperbolic trigonometry if the sides${\displaystyle a\text{,}b\text{,}c}$ are replaced by${\displaystyle ia\text{,}ib\text{,}ic}$ in the formulas for spherical geometry. Here${\displaystyle i}$ is the square root of -1. This does not constitute a proof but does mean that further formulas in spherical trigonometry derived from those formulas can be transformed into formulas for hyperbolic trigonometry by making those substitutions. This is so since the circular trigonometric functions areanalytic functions and the trigonometric identities are valid for complex arguments. In particular.${\displaystyle \cos (ia)=\cosh (a)}$,${\displaystyle \sin (ia)=i\sinh (a)}$,${\displaystyle \tan (ia)=i\tanh (a)}$, and${\displaystyle \cot (ia)=-i\coth (a)}$. A hyperbolic triangle in which a b A B are known but c C are not known cannot be solved straightforwardly using the two laws of cosines and the law of sines but can be solved using theNapier's analogies. The methods forsolving spherical triangles can then be applied to hyperbolic triangles.

• Pair of pants (mathematics)
• Spherical trigonometry
• Triangle group

For hyperbolic trigonometry:

• Angle of parallelism
• Hyperbolic law of cosines
• Hyperbolic law of sines
• Lambert quadrilateral
• Saccheri quadrilateral

• Svetlana Katok (1992)Fuchsian Groups,University of Chicago PressISBN 0-226-42583-5

• Hyperbolic geometry
• Types of triangles

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