spherical trigonometry

Spherical triangle

Spherical trigonometry is a branch of spherical geometry which deals with polygons (especially triangles) on the sphere and the relationships between the sides and the angles. This is of great importance for calculations in astronomy and earth-surface, orbital and space navigation.

History

Spherical trigonometry was studied by early Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem.^[1]

In the 10th century, Abū al-Wafā' al-Būzjānī established the angle addition formulas, e.g., sin(a + b), and discovered the sine formula 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 refer not to the vertices of the triangle but to the interior angles at those vertices—ignore the Greek letters.)

Al-Jayyani (989-1079), an Arabic mathematician in Islamic Iberian Peninsula, wrote what some consider the first treatise on spherical trigonometry, circa 1060, entitled The book of unknown arcs of a sphere,^[2] 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.^[3] Al-Jayyani's book "contains formulae for right-angle triangles, the general law of sines and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus.^[2]

In the 13th century, Iranian mathematician Nasīr al-Dīn al-Tūsī 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.^[4] 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 are great circles, i.e. circles whose center coincide with the center of the sphere. For example, meridians and the equator are great circles on the Earth, while non-equatorial lines of latitude are small circles. As with a line segment in a plane, an arc of a great circle (subtending less than 180°) on a sphere is the shortest path lying on the sphere between its two endpoints. Great circles are special cases of the concept of a geodesic.

An area on the sphere bounded by arcs of great circles is called a spherical polygon. Note that, unlike the case on a plane, spherical "biangles" (two-sided analogs to triangle) are possible (such as a slice cut out of an orange). Such a polygon is also called a lune.

The sides of these polygons are specified not by their lengths, but by the angles at the sphere's center subtended to the endpoints of the sides. Note that this arc angle, measured in radians, when multiplied by the sphere's radius equals the arc length. (In the special case of polygons on the surface of a sphere of radius one, the arc length of any side equals its subtended angle.)

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 spherical triangles is always larger than the sum of the angles of plane triangles, which is exactly 180°. The amount E by which the sum of the angles exceeds 180° is called spherical excess:

E = α + β + γ − π,

where α, β and γ denote the angles. Girard's theorem, named after the 16th century French mathematician Albert Girard (earlier discovered but not published by the English mathematician Thomas Harriot), states that this surplus determines the surface area of any spherical triangle:

where R is the radius of the sphere.

The analogous result holds for hyperbolic triangles, with "excess" replaced by "defect"; these are both special cases of the Gauss-Bonnet theorem.

It follows from here that there are no non-trivial 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:

Napier's Circle shows the relations of parts of a right spherical triangle

Napier's pentagon (also known as Napier's circle) is a mnemonic aid 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 sines of the two angles written opposed to it

As an example, starting with the angle , we can obtain the formula:

See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation.

Identities

Spherical triangle solved by the law of cosines.

Spherical triangles satisfy a spherical law of cosines

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 area limit.

They also satisfy an analogue of the law of sines

A more thorough list of identities is available here

See also

• Air navigation
• Spherical geometry
• Spherical distance
• Spherical polyhedron
• Celestial navigation
• SIGI

References

• Isaac Todhunter: Spherical Trigonometry: For the Use of Colleges and Schools. Macmillan & Co. 1863 (complete online version (Google Books))

External links

• Wolfram's mathworld: Spherical Trigonometry a more thorough list of identities, with some derivation
• Wolfram's mathworld: Spherical Triangle nice applet
• Intro to Spherical Trig. Includes discussion of The Napier circle and Napier's rules
• Spherical Trigonometry — for the use of colleges and schools by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by Cornell University Library.
• A Visual Proof of Girard's Theorem by Okay Arik, the Wolfram Demonstrations Project.