All Topics  
Spherical trigonometry

 

   Email Print
   Bookmark   Link






 

Spherical trigonometry



 
 
Spherical trigonometry is a part of spherical geometry
Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
 that deals with polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s (especially triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s) on the sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 and explains how to find relations between the involved angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s. This is of great importance for calculations in astronomy
Astronomy

Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
 and earth-surface and orbital and space navigation
Navigation

Navigation is the process of reading, and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks....
.

rical trigonometry was dealt with by early Greek mathematicians
Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics written in Greek language, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean....
 such as Menelaus of Alexandria
Menelaus of Alexandria

Menelaus of Alexandria, Egypt was a Greeks mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines....
 who wrote a book that dealt with spherical trigonometry called Sphaerica and developed Menelaus' theorem
Menelaus' theorem

Menelaus' theorem, attributed to Menelaus of Alexandria, is a theorem about triangles in plane geometry. Given points A, B, C that form triangle ABC, and points D, E, F that lie on lines BC, AC, AB, then the theorem states that D, E, F are collinear if and only if:...
.

In the 10th century, Abu al-Wafa' al-Buzjani established the angle addition identities, e.g., sin(a + b), and discovered the sine formula
Law of sines

The law of sines , in trigonometry, is a statement about any triangle in a plane. Where the sides of the triangle are a, b and c and the angles opposite those sides are A, B and C, then the law of sines states equality of the first three quantities below:...
 for spherical trigonometry:

Here, a, b, and c are the angles at the centre of the sphere subtended by the three sides of the triangle, and A, B, and C are the angles between the sides, where angle A is opposite the side which subtends angle a, etc.






Discussion
Ask a question about 'Spherical trigonometry'
Start a new discussion about 'Spherical trigonometry'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Spherical trigonometry is a part of spherical geometry
Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
 that deals with polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
s (especially triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s) on the sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 and explains how to find relations between the involved angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
s. This is of great importance for calculations in astronomy
Astronomy

Astronomy is the science of Astronomical object and Phenomenon that originate outside the Earth's atmosphere . It is concerned with the evolution, physics, chemistry, meteorology, and motion of celestial objects, as well as the physical cosmology....
 and earth-surface and orbital and space navigation
Navigation

Navigation is the process of reading, and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks....
.

History

Spherical trigonometry was dealt with by early Greek mathematicians
Greek mathematics

Greek mathematics, as that term is used in this article, is the mathematics written in Greek language, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean....
 such as Menelaus of Alexandria
Menelaus of Alexandria

Menelaus of Alexandria, Egypt was a Greeks mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines....
 who wrote a book that dealt with spherical trigonometry called Sphaerica and developed Menelaus' theorem
Menelaus' theorem

Menelaus' theorem, attributed to Menelaus of Alexandria, is a theorem about triangles in plane geometry. Given points A, B, C that form triangle ABC, and points D, E, F that lie on lines BC, AC, AB, then the theorem states that D, E, F are collinear if and only if:...
.

In the 10th century, Abu al-Wafa' al-Buzjani established the angle addition identities, e.g., sin(a + b), and discovered the sine formula
Law of sines

The law of sines , in trigonometry, is a statement about any triangle in a plane. Where the sides of the triangle are a, b and c and the angles opposite those sides are A, B and C, then the law of sines states equality of the first three quantities below:...
 for spherical trigonometry:

Here, a, b, and c are the angles at the centre of the sphere subtended by the three sides of the triangle, and A, B, and C are the angles between the sides, where angle A is opposite the side which subtends angle a, etc. (In the diagram A, B, C should be taken to refer not to the vertices of the triangle but to the interior angles at those vertices—ignore the Greek letters.)

Al-Jayyani
Al-Jayyani

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani, shortened to Al-Jayyani was an Arab Islamic mathematics from Al-Andalus . Al-Jayyani wrote important commentaries on Euclid's Euclid's Elements and he wrote the first treatise on spherical trigonometry in its modern form....
 (989-1079), an Arabic mathematician
Islamic mathematics

Mathematics in medieval Islam or sometimes referred to as Islamic mathematics is a term used in the history of mathematics that refers to the mathematics developed in the Muslim world between 622 and 1600, in the part of the world where Islam was the dominant religion....
 in Islamic Spain
Al-Andalus

Al-Andalus was the Arabic name given to the parts of the Iberian Peninsula governed by Arab Muslims, at various times in the period between 711 and 1492....
, wrote what some consider the first treatise on spherical trigonometry, circa 1060, entitled The book of unknown arcs of a sphere, in which spherical trigonometry was brought into its modern form. E. S. Kennedy points out that while it was possible in ancient mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. Al-Jayyani's book "contains formulae for right-angle triangles
Special right triangles

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist....
, the general law of sines
Law of sines

The law of sines , in trigonometry, is a statement about any triangle in a plane. Where the sides of the triangle are a, b and c and the angles opposite those sides are A, B and C, then the law of sines states equality of the first three quantities below:...
 and the solution of a spherical triangle by means of the polar triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
." This treatise later had a "strong influence on European mathematics", and his "definition of ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
s as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus
Regiomontanus

Johannes M?ller von K?nigsberg , known by his Latin pseudonym Regiomontanus, was an important Germany mathematician, astronomer and astrologer....
.

In the 13th century, Iranian mathematician Nasir al-Din al-Tusi was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he further developed spherical trigonometry, bringing it to its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry.

Lines on a sphere

On the surface of a sphere, the closest analogue to straight lines
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 are great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
s, i.e. circles whose center coincide with the center of the sphere. For example, meridian
Meridian (geography)

A meridian is an imaginary arc on the Earth's surface from the North Pole to the South Pole that connects all locations running along it with a given longitude....
s and the equator
Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
 are great circles on the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
, while non-equatorial lines of latitude are not great circles. As with a line segment in a plane
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
, an arc
Arc (geometry)

In geometry, an arc is a closed set segment of a differentiable curve in the two-dimensional manifold; for example, a circular arc is a segment of the circumference of a circle....
 of a great circle (subtending less than 180°) on a sphere is the shortest path lying on the sphere between its two endpoints
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
. Great circles are special cases of the concept of a geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
.

An area on the sphere which is bounded by arcs of great circle
Great circle

A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere....
s is called a spherical polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
. Note that, unlike the case on a plane, spherical "biangles
Digon

In geometry a digon is a Degeneracy polygon with two sides and two Vertex .A digon must be Regular polygon because its two edges are the same length....
" (two-sided analogs to triangle) are possible (such as a slice cut out of an orange). Such a polygon is also called a lune
Lune (mathematics)

A lune is either of two figures, both shaped roughly like a crescent Moon. The word "lune" derives from luna, the Latin language word for Moon....
.

The side
Side

Side is one of the best-known classical sites in Turkey, and was an ancient harbour whose name meant pomegranate. Side is a resort town on the southern coast of Turkey, near the villages of Manavgat and Selimiye , 75 km from Antalya) in the Antalya Province....
s of these polygons are most conveniently specified not by their length but by the angle under which its endpoints appear when looked at from the sphere's center. Note that this arc angle, measured in radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s, when multiplied by the sphere's radius
RADIUS

Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
 equals the arc length.

Hence, a spherical triangle is specified as usual by its corner angles and its sides, but the sides are given not by their length, but by their arc angle.

The sum of the vertex angles of a spherical triangle is always larger than the 180° found in every planar triangle. The amount E by which the sum of the angles exceeds 180° is called the spherical excess: where a, ß and ? denote the angles. Girard's theorem states that this surplus determines the surface area of any spherical triangle: where R is the radius of the sphere.

It follows from here that there are no similar triangles (triangles with equal angles but different side lengths and area) on a sphere. In the special case of a sphere of radius 1, the area simply equals the excess angle: A = E. One can also use Girard's formula to obtain the discrete Gauss-Bonnet theorem.

To solve a geometric problem on the sphere, one dissects the relevant figure into right spherical triangles (i.e.: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon:

Neper's Circle
Napier
John Napier

John Napier of Merchistoun - also signed as Neper, Nepair - named Marvellous Merchiston, was a Scotland mathematics, physicist, astronomer/astrologer and 8th Laird of Merchistoun, son of Sir Archibald Napier of Merchiston....
's pentagon (also known as Napier's circle) is a mnemonic aid
Mnemonic

A mnemonic device is a memory aid. Commonly met mnemonics are often verbal, something such as a very short poem or a special word used to help a person remember something, particularly lists, but may be visual, kinesthetic or auditory....
 that helps to find all relations between the angles in a right spherical triangle.

Write the six angles of the triangle (three vertex angles, three arc angles) in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles adjacent to it by their complement to 90° (i.e. replace, say, a by 90° − a). The five numbers that you now have on your paper form Napier's Pentagon (or Napier's Circle). For them, it holds that the cosine of each angle is equal to:
  • the product of the cotangents of the angles written next to it
  • the product of the sine
    Siné

    Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
    s of the two angles written opposed to it
See also the Haversine formula
Haversine formula

The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes....
, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation.

Identities

Spherical triangles satisfy a spherical law of cosines
Law of cosines (spherical)

In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry....


The identity may be derived by considering the triangles formed by the tangent lines to the spherical triangle subtending angle C and using the plane law of cosines. Moreover, it reduces to the plane law in the small angle limit.

They also satisfy an analogue of the law of sines
Law of sines

The law of sines , in trigonometry, is a statement about any triangle in a plane. Where the sides of the triangle are a, b and c and the angles opposite those sides are A, B and C, then the law of sines states equality of the first three quantities below:...


A more thorough list of identities is available

See also

  • Air navigation
    Air navigation

    The principles of air navigation are the same for all aircraft, big or small. Air navigation involves successfully piloting an aircraft from place to place without getting lost, breaking the laws applying to aircraft, or endangering the safety of those on board or on the ground....
  • Spherical geometry
    Spherical geometry

    Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a non-Euclidean geometry. Two practical applications of the principles of spherical geometry are navigation and astronomy....
  • Spherical distance
    Great-circle distance

    The great-circle distance is the shortest distance between any two Point s on the surface of a sphere measured along a path on the surface of the sphere ....
  • Spherical polyhedron
    Spherical polyhedron

    In mathematics, the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tessellation or spherical polyhedron....
  • Celestial navigation
    Celestial navigation

    Celestial navigation, also known as astronavigation, is a position fixing technique that was devised to help sailors cross the featureless oceans without having to rely on dead reckoning to enable them to strike land....
  • SIGI
    Sigi

    In the V?lsung cycle, Sigi is the ancestor of the V?lsung lineage.In the V?lsunga saga , he is said to be the Sons of Odin....


External links

  • a more thorough list of identities, with some derivation
  • nice applet
  • Includes discussion of The Napier circle and Napier's rules
  • by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by .
  • by Okay Arik, the Wolfram Demonstrations Project
    Wolfram Demonstrations Project

    The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
    .