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

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.

Activity 1.6.1 demonstrates 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.

Answer

Figure 1.6.3. A spherical triangle with three right angles.