Spherical trigonometry
Spherical trigonometry is a part of spherical geometry that deals with
polygons 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.
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 . As lines on a plane, great circles on a sphere are the closest connection of two points .
An area on the sphere which is bounded by arcs of great circles is called a spherical
polygon.
Encyclopedia
Spherical trigonometry is a part of spherical geometry that deals with
polygons 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.
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 . As lines on a plane, great circles on a sphere are the closest connection of two points .
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" are possible .
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 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 because one can then use Napier's pentagon:
Napier's pentagon 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 . Then cross out the 90° corner angle and replace the arc angles adjacent to it by their complement to 90° . The five numbers that you now have on your paper form Napier's Pentagon . 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.
Identities
Spherical triangles satisfy a
spherical law of cosinesThe 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 sinesA more thorough list of identities is available
See also
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 .
