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Hyperbolic triangle

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Title: Hyperbolic triangle  
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Hyperbolic triangle

A hyperbolic triangle embedded in a saddle-shaped surface

In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points 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.

A tiling of the hyperbolic plane with hyperbolic triangles – the order-7 triangular tiling.

Contents

  • Definition 1
  • Properties 2
  • Triangles with ideal vertices 3
  • Standardized Gaussian curvature 4
  • Trigonometry 5
    • Trigonometry of right triangles 5.1
      • Area 5.1.1
      • Angle of parallelism 5.1.2
    • General trigonometry 5.2
  • See also 6
  • References 7
  • Further reading 8

Definition

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

Properties

The relations among the angles and sides are analogous to those of spherical trigonometry

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 two right angles. This is contrasted to Euclidean triangles where this sum is always equal to the straight angle, as well as to spherical triangles where this sum is greater. The difference between the measure of two right angles. and the sum of the measures of a triangle's angles is called the defect of the triangle.

As in Euclidean geometry each hyperbolic triangle has an inscribed circle. But if its vertices lie on an horocycle or hypercycle, the triangle has no circumscribed circle.

As in spherical geometry the only similar triangles are congruent triangles.

Triangles with ideal vertices

Three ideal triangles in the Poincaré disk model

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 asymptotic (i.e. the distance between them vanishes 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 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 a 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). This choice for this length scale makes formulas simpler.[2]

In terms of the (constant and negative) Gaussian curvature K of a hyperbolic plane this unit of length is given by

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

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 R:

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

This theorem, first proven by Johann Heinrich Lambert,[3] is related to Girard's theorem in spherical geometry.

Trigonometry

In all the formulas stated below the sides a, b, and c must be measured in a unit so that the Gaussian curvature K of the plane is −1. In other words, R 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 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.
\sin A=\frac{\textrm{sinh(opposite)}}{\textrm{sinh(hypotenuse)}}=\frac{\sinh a}{\,\sinh c\,}.\,
  • The cosine of angle A is the ratio of the hyperbolic tangent of the adjacent leg to the hyperbolic tangent of the hypotenuse.
\cos A=\frac{\textrm{tanh(adjacent)}}{\textrm{tanh(hypotenuse)}}=\frac{\tanh b}{\,\tanh c\,}.\,
  • The tangent of angle A is the ratio of the hyperbolic tangent of the opposite leg to the hyperbolic sine of the adjacent leg.
\tan A=\frac{\textrm{tanh(opposite)}}{\textrm{sinh(adjacent)}} = \frac{\tanh a}{\,\sinh b\,}.
  • The hyperbolic cosine of the hypotenuse is the product of hyperbolic cosine of the adjacent leg and the hyperbolic cosine of the opposite leg.
\textrm{cosh(hypotenuse)}= \textrm{cosh(adjacent)} \textrm{cosh(opposite)}.
  • The hyperbolic cosine of the adjacent leg to angle A is the ratio of the cosine of angle B to the sine of angle A.
\textrm{cosh(adjacent)}= \frac{\cos B}{\sin A}.
  • The hyperbolic cosine of the hypotenuse is the ratio of the product of the cosines of the angles to the product of their sines.[4]
\textrm{cosh(hypotenuse)}= \frac{\cos A \cos B}{\sin A\sin B}.

Area

The area of a right angled triangle is :

\textrm{Area} = \frac{\pi}{2} - \angle A - \angle B

also

\textrm{Area}= 2 \arctan (\tanh (\frac{a}{2})\tanh (\frac{b}{2}) )[5]

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{tanh}(\infty )= 1 , resulting in \cos A= \textrm{tanh(adjacent)}.

General trigonometry

Whether C is a right angle or not, the following relationships hold: The hyperbolic law of cosines is as follows:

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

Its dual is

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

There is also a law of sines:

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

and a four-parts formula:

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

See also

For hyperbolic trigonometry:

References

  1. ^ Stothers, Wilson (2000), Hyperbolic geometry,  , interactive instructional website
  2. ^ Needham, Tristan (1998). Visual Complex Analysis. Oxford University Press. p. 270.  
  3. ^ 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.
  4. ^ Martin, George E. (1998). The foundations of geometry and the non-Euclidean plane (Corrected 4. print. ed.). New York, NY: Springer. p. 433.  
  5. ^ "Area of a right angled hyperbolic triangle as function of side lengths". Mathematics  

Further reading

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