Spherical Trigonometry - Lines and Angles On A Sphere

Lines and Angles On A Sphere

On the surface of a sphere, the closest analogue to straight lines are great circles, i.e. circles whose centers coincide with the center of the sphere. For example, simplifying the shape of the Earth (the geoid) to a sphere, the meridians and the equator are great circles on its surface, 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:

where α, β and γ denote the angles in degrees. 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 and is the spherical excess measured in radians. From this and the area formula for a sphere it follows that the sum of the angles of a spherical triangle is .

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.