Examples of oblique spherical triangle

Answer 1

One example of an oblique spherical triangle is obtained by choosing any two points on the surface of the earth, perhaps two cities. (The earth is assumed to be a perfect sphere.) Now connect these two points with the shortest path possible on the surface of the earth. On a plane, the shortest path is a straight line, but a straight line connecting our two cities would pass below the surface of the earth. Instead, pass a plane through the center of the earth and the two cities. The intersection of this plane with the sphere is a great circle. (If you have a globe, stretch a string taught while it touches both cities, and the string will follow the great circle.) There are actually two ways to connect the cities... the "short" way and the "long" way. (The "long" way goes almost completely around the earth if the cities are close together.) Choose either one. Now, connect city one to the north pole with another great circle. This great circle is a meridian line (a line with the same longitude at all points). Again, you can choose the short or the long way around. (Note, if you choose the long way around, then the longitude of the line shifts abruptly at the south pole by 180 degrees.) Also connect city two to the north pole with a great circle. And you are done.

The three lines you have chosen, make a spherical triangle.

Why bother to make this triangle? One practical application is to use spherical trig with this spherical triangle. Given the latitude and longitude of the two points, one can easily compute the distance (along the surface of the earth) between the two cities.