Friday, January 19, 2007

The shortest (flight) path problem

While on the unbelievably long India flight, I was thinking about flight paths that planes take when they travel around the world. I stuck upon this easy way of finding the shortest path between any 2 cities in the world.

Most people believe that the shortest distance between any 2 cities in the world is via the path obtained by drawing a straight line between them on a map. Take for instance Seattle and Paris. Both lie roughly on the 48th latitude. Seattle around 120 E longitude, while Paris roughly near 0 i.e. Greenwich Meridian. So the fastest way to go from Seattle to Paris is to simply go east along the 48th latitude till we hit Paris. Right? Wrong..... The truth is, that unlike a map, the earth is not flat. It is a sphere (It is actually a geoid, but for the sake of argument here we will assume it is a nice symmetrical sphere).

So what is the shortest distance between any two points on the surface of a sphere? Here is a simple way of finding it out:
Draw a circle, centered at the center of the earth and passing through these two cities. The smaller arc of the circle joining the two points is the required shortest path. The 3d symmetry of a sphere dictates that this should be the right answer and I am not gonna take the pains of proving it :). It is true at the poles and true at the equator and you could possibly prove by induction that it is true every point in between.

Thus for any 2 cities in the Northern Hemisphere, the shortest flight path curves northwards and vice versa for cities in the Southern Hemisphere. For cities on symmetrically opposite longitudes, e.g. Seattle (~120 W) and Dubai (~60 E), the shortest path passes through one of the poles (in this case, the North Pole).