Spherical trigonometry


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.
The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Isaac Todhunter's textbook Spherical trigonometry for the use of colleges and Schools.
Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.

Preliminaries

Spherical polygons

A spherical polygon is a polygon on the surface of the sphere. Its sides are arcs of great circles—the spherical geometry equivalent of line segments in plane geometry.
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also called digons or bi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.
One spherical polygon with interesting properties is the pentagramma mirificum, a 5-sided spherical star polygon with a right angle at every vertex.
From this point in the article, discussion will be restricted to spherical triangles, referred to simply as triangles.

Notation

  • Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters,, and.
  • Sides are denoted by lower-case letters:,, and. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent.
  • The angle may be regarded either as the dihedral angle between the two planes that intersect the sphere at the vertex, or, equivalently, as the angle between the tangents of the great circle arcs where they meet at the vertex.
  • Angles are expressed in radians. The angles of proper spherical triangles are less than, so that.
In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly radians.
  • Sides are also expressed in radians. A side is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention, the sides of proper spherical triangles are less than, so that.
  • The sphere's radius is taken as unity. For specific practical problems on a sphere of radius the measured lengths of the sides must be divided by before using the identities given below. Likewise, after a calculation on the unit sphere the sides,, and must be multiplied by .

    Polar triangles

The polar triangle associated with a triangle is defined as follows. Consider the great circle that contains the side . This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as is termed the pole of and it is denoted by. The points and are defined similarly.
The triangle is the polar triangle corresponding to triangle . The angles and sides of the polar triangle are
given by
Therefore, if any identity is proved for then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.
If the matrix has the positions,, and as its columns then the rows of the matrix inverse, if normalized to unit length, are the positions,, and. In particular, when is the polar triangle of then is the polar triangle of.

Cosine rules and sine rules

Cosine rules

The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
These identities generalize the cosine rule of plane trigonometry, to which they are asymptotically equivalent
in the limit of small interior angles.

Sine rules

The spherical law of sines is given by the formula
These identities approximate the sine rule of plane trigonometry when the sides are much smaller than the radius of the sphere.

Derivation of the cosine rule

The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule. He also gives a derivation using simple coordinate geometry and the planar cosine rule. The approach outlined here uses simpler vector methods.
Consider three unit vectors drawn from the origin to the vertices of the triangle. The arc subtends an angle of magnitude at the centre and therefore. Introduce a Cartesian basis with along the -axis and in the -plane making an angle with the -axis. The vector projects to in the -plane and the angle between and the -axis is. Therefore, the three vectors have components:
The scalar product in terms of the components is
Equating the two expressions for the scalar product gives
This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
The other cosine rules are obtained by cyclic permutations.

Derivation of the sine rule

This derivation is given in Todhunter,. From the identity and the explicit expression for given immediately above
Since the right hand side is invariant under a cyclic permutation of,, and the spherical sine rule follows immediately.

Alternative derivations

There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter gives two proofs of the cosine rule and two proofs of the sine rule. The page on Spherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy and spherical astronomy give different proofs and the online resources of MathWorld provide yet more. There are even more exotic derivations, such as that of Banerjee who derives the formulae using the linear algebra of projection matrices and also quotes methods in differential geometry and the group theory of rotations.
The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule. The scalar triple product, evaluates to in the basis shown. Similarly, in a basis oriented with the -axis along, the triple product, evaluates to. Therefore, the invariance of the triple product under cyclic permutations gives which is the first of the sine rules. See curved variations of the law of sines to see details of this derivation.

Differential variations

When any three of the differentials da, db, dc, dA, dB, dC are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination:

Identities

Supplemental cosine rules

Applying the cosine rules to the polar triangle gives, i.e. replacing by, by etc.,

Cotangent four-part formulae

The six parts of a triangle may be written in cyclic order as. The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example or ). In such a set there are inner and outer parts: for example in the set the inner angle is, the inner side is, the outer angle is, the outer side is. The cotangent rule may be written as
and the six possible equations are :
To prove the first formula start from the first cosine rule and on the right-hand side substitute for from the third cosine rule:
The result follows on dividing by. Similar techniques
with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.

Half-angle and half-side formulae

With and
Another twelve identities follow by cyclic permutation.
The proof of the first formula starts from the identity using the cosine rule to express in terms of the sides and replacing the sum of two cosines by a product. The second formula starts from the identity the third is a quotient and the remainder follow by applying the results to the polar triangle.

Delambre analogies

The Delambre analogies were published independently by Delambre, Gauss, and Mollweide in 1807–1809.
Another eight identities follow by cyclic permutation.
Proved by expanding the numerators and using the half angle formulae.

Napier's analogies

Another eight identities follow by cyclic permutation.
These identities follow by division of the Delambre formulae.
Taking quotients of these yields the law of tangents, first stated by Persian mathematician Nasir al-Din al-Tusi,