PHY115: Professional Skills in Physics and Astronomy

Positional Astronomy: Session 3

Spherical Triangles

We discussed in the first session that geometry on the surface of a sphere differs from geometry on a plane. Of particular importance is the difference in trigonometry. Just as you can define a triangle on a planar surface, you can define a triangle on the surface of a sphere.

A planar triangle is the shape which connects three points by the shortest route (along straight lines). In the same way, a spherical triangle connects three points by the shortest route. Since we are now working on the surface of a sphere, the sides of a spherical triangle are no longer straight lines, but follow great circles, which we defined in the first session.

The complete definition of a spherical triangle is that it must fulfil all of the following conditions:

• each side is a part of a great circle,
• any two sides together are longer than the third side,
• each angle is less than 180°,
• the sum of the three angles is greater than 180°.
Figure 1: A spherical triangle on the surface of the Earth.
You can immediately see that many of the rules you know for planar triangles no longer apply to spherical triangles. Consider the spherical triangle shown in the figure to the right. It connects the points PAB. To follow the sides of this triangle you would start at the north pole, and travel south to the equator. There you would turn left by 90° and walk a quarter of the way around the equator. At that point you'd turn left through 90° again and walk back to the pole. To get back to the way you were originally facing you'd have to turn left through 90° again. The sum of the three angles in the spherical triangle PAB is 270°!

© 2010 Stuart Littlefair Contact Me