Hyperbolic triangle

Last updated October 16, 2024

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.

Definition

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

Properties

Hyperbolic triangles have some properties that are analogous to those of triangles 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.

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 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.
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 (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 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.

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). This choice for this length scale makes formulas simpler.^[2]

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

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

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 absolute length, a unit so that the Gaussian curvature $K$ of the plane is −1. In other words, the quantity $R$ 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 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 {\textrm {sinh(opposite)}}{\textrm {sinh(hypotenuse)}}}={\frac {\sinh a}{\,\sinh c\,}}.\,

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

{\textrm {cosh(adjacent)}}={\frac {\cos B}{\sin A}}

.

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

{\textrm {cosh(hypotenuse)}}={\textrm {cosh(adjacent)}}{\textrm {cosh(opposite)}}

.

The hyperbolic cosine of the hypotenuse is also the product of the cosines of the angles divided by the product of their sines.^[4]

{\textrm {cosh(hypotenuse)}}={\frac {\cos A\cos B}{\sin A\sin B}}=\cot A\cot B

Relations between angles

We also have the following equations:^[5]

\cos A=\cosh a\sin B

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

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

\cos B=\cosh b\sin A

\cosh c=\cot A\cot 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}}))

^{[ citation needed ]}^[6]

The area for any other triangle is:

{\textrm {Area}}={\pi }-\angle A-\angle B-\angle C

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 $\cos A={\textrm {tanh(adjacent)}}$ .

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 {{\textrm {tanh}}({\frac {1}{2}}s)}{{\textrm {tanh}}(s)}}

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

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 theorem 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

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

Related Research Articles

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points $(cos t, sin t)$ form a circle with a unit radius, the points $(cosh t, sinh t)$ form the right half of the unit hyperbola. Also, similarly to how the derivatives of $sin(t)$ and $cos(t)$ are $cos(t)$ and $-sin(t)$ respectively, the derivatives of $sinh(t)$ and $cosh(t)$ are $cosh(t)$ and $+sinh(t)$ respectively.

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, $where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles, while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals; The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data and the technique gives two possible values for the enclosed angle.$

<span class="mw-page-title-main">Hypotenuse</span> Longest side of a right-angled triangle, the side opposite of the right angle

In geometry, a hypotenuse is the side of a right triangle opposite the right angle. It is the longest side of any such triangle; the two other shorter sides of such a triangle are called catheti or legs. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two legs. Mathematically, this can be written as $, where a is the length of one leg, b is the length of another leg, and c is the length of the hypotenuse.$

In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.

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 great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.

<span class="mw-page-title-main">Bipolar coordinates</span> 2-dimensional orthogonal coordinate system based on Apollonian circles

Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There is also a third system, based on two poles.

<span class="mw-page-title-main">Angle of parallelism</span> An angle in certain right triangles in the hyperbolic plane

In hyperbolic geometry, angle of parallelism $is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.$

In geometry, a Lambert quadrilateral, is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted as and .$

Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios such as sine.

In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides $and opposite respective angles and, the law of cosines states:$

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 trigonometry. It can also be related to the relativistic velocity addition formula.

Solution of triangles is the main trigonometric problem of finding the characteristics of a triangle, when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.

<span class="mw-page-title-main">Pythagorean theorem</span> Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

<span class="mw-page-title-main">Law of cotangents</span> Trigonometric identity relating the sides and angles of a triangle

In trigonometry, the law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.

In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.

References

↑ Stothers, Wilson (2000), Hyperbolic geometry, University of Glasgow , interactive instructional website
↑ Needham, Tristan (1998). Visual Complex Analysis. Oxford University Press. p. 270. ISBN 9780198534464.
↑ Ratcliffe, John (2006). Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. Vol. 149. Springer. p. 99. ISBN 9780387331973. 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.
↑ Martin, George E. (1998). The foundations of geometry and the non-Euclidean plane (Corrected 4. print. ed.). New York, NY: Springer. p. 433. ISBN 0-387-90694-0.
↑ Smogorzhevski, A.S. Lobachevskian geometry. Moscow 1982: Mir Publishers. p. 63.{{cite book}}: CS1 maint: location (link)
↑ "Area of a right angled hyperbolic triangle as function of side lengths". Stack Exchange Mathematics. Retrieved 11 October 2015.