Spherical Geometry I

Section 1.6 Spherical Geometry I

Activity 1.6.1. The explorer.

An explorer set off from camp and walked 1 mile south, discovering there some interesting tracks. The explorer followed the tracks for 1 mile to the west, then returned to camp by going 1 mile north. Where was the camp?

Hint

It's a bear! What color was the bear?

Solution 1

At the North Pole! (It's a polar bear!)

Figure 1.6.1. The path of the explorer.

Can you find another solution?

Solution 2

Near the South Pole! (It's a penguin!)

Figure 1.6.2. An alternate path for the explorer.

Activity 1.6.2. Spherical Triangles.

The solutions to Activity 1.6.1 demonstrate that geometry on a sphere (the earth) is not the same as on a plane. Notice that the triangle in the first solution is equilateral, but its angles are not all equal! Another surprising example of a spherical triangle is shown in Figure 1.6.3, which shows a spherical triangle with three right angles. Try to draw such a triangle yourself before looking at the answer. ¹

As discussed in Section 7.6, the “triangle” in Figure 1.6.1 isn't quite right, since lines of latitude are not great circles, and therefore do not count as lines in spherical geometry. However, the triangle in Figure 1.6.3 is indeed a spherical triangle, since the equator is a great circle.

Answer

Figure 1.6.3. A spherical triangle with three right angles.