Spherical Triangles

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Fig 2a shows three planes, all at right angles to each another and centered at the origin. In 2b, the planes intersect with a sphere, also centered at the origin. The intersection of each plane with the sphere forms a great circle. So each great circle lies in its own plane (2c). Where the great circles intersect, they form a three sided figure that looks like a triangle, drawn on the surface of the sphere. There are eight such triangles and the one facing is marked ABC. The edges AB, BC, and CA are all geodesic lines – ie arcs of great circles. A triangle on the surface of a sphere, whose edges are all geodesic lines is called a spherical triangle.

Fig 2a. What is angle B of the spherical triangle? The spherical angle between two great circles is measured by the angle between the planes on which they lie. So B must be 90°. Since all three great circles are at right angles to each other, if follows that A = 90° and C = 90°. The sum of the angles in spherical triangle ABC is therefore 270°. Thus the three angles of a spherical triangle are together greater than two right angles – an important difference between spherical triangles and the flat triangles of Euclidian geometry.

The sides of spherical triangles are always measured in degrees or radians.

So the side AB and the side AC are both 90° or pi over -two radians. Thus a spherical triangle does not primarily represent distances between its points, but rather the differences in direction of its points, when seen from the center of a sphere. The side AB can be thought of as the angle through which they eye must turn to get from A to B, when standing at the center of the sphere.


Above is another example. The spherical angle PQR between geodesic lines PQ and RQ is the angle PoR between their intersecting planes.

PoR = 90° - PoZ
EoZ = 90° - PoZ

So the angle between geodesic lines PQ and RQ is also the angle between the axes of the great circles on which they lie, which is the angle EoZ.