103 Trigonometry Problems

56 103 Trigonometry Problems
how can we compute the length of the shortest path along the surface connecting
the two places? Let C denote this path. Intuitively, it is not difficult to see that the
points on C all lie on a plane. This fact is not that easy to prove, however, so let’s
just accept it. Let P denote the plane that contains C. It is clear that the intersection
of P and Earth’s surface is a circle. Consequently, we conclude that C is an arc. For
two fixed points, we can draw many circles passing through these two points (Figure
1.55). It is not difficult to see that as the radii of the circles increase, the lengths of the
minor arcs connecting the points decrease. (When the radius of the circle approaches
infinity, the circle becomes a line and the minor arc connecting the points becomes
the segment connecting the points.)
Figure 1.55.
For two points A and B on the surface of Earth, the largest circle passing through
A and B on Earth is the circle that is centered at Earth’s center. Such circles are
called great circles. For example, the equator is a great circle, and all points with
fixed longitudes form great semicircles. We encourage the reader to choose pairs of
arbitrary points on a globe and draw great circles passing through the chosen points.
The reader might then have a better idea why many intercontinental airplanes fly at
high latitudes.
B
A
O
Figure 1.56.