Spherical trigonometry

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Spherical triangle

Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. This is of great importance for calculations in astronomy and earth-surface and orbital and space navigation.

Al-Jayyani, an Arabic scholar in Moorish Spain, wrote the first treatise on spherical trigonometry in 1060 AD.

1 Lines on a Sphere
2 Identities
3 See also
4 External links

[edit] 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). As lines on a plane, great circles on a sphere are the closest connection of two points (if you constrain yourself to lines on the sphere). (cf. geodesic)

An area on the sphere which is 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 (think about peeling an orange).

The sides 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 radians, and multiplied by the sphere's radius, is 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.

Remarkably, the sum of the vertex angles of a spherical triangle is always larger than the 180° found in every planar triangle. The amount by which the sum of the angles exceeds 180° is called the spherical excess E: E = α + β + γ − 180°. This surplus determines the surface area of any spherical triangle. To determine this, the spherical excess must be expressed in radians; the surface area A is then given in terms of the sphere's radius R by the expression:

A = R² · E. From this formula, which is an application of the Gauss-Bonnet theorem, it becomes obvious 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.

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 to easily find all relations between the angles in a right spherical triangle:

Write the six angles of the triangle 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

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

[edit] Identities

Spherical triangles satisfy a spherical law of cosines

$\cos c= \cos a \cos b + \sin a \sin b \cos C \!$

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

$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.$