Wednesday, June 25, 2008

Wednesday Math, Vol. 29: Distance on a sphere

(By request from friend of the blog DistributorCap.)

It's well known that the shortest distance between two points is a line segment. That's true in two dimensions or three dimensions. To get all technical and math-y, it's true in more dimensions as well. But if we have two points on the surface of a sphere, we can't use the straight line between them, because the sphere gets in the way. What's does "shortest distance" mean then?

Two points determine a unique line in three dimensions, but infinitely many planes contain that line. If the two points are also on the surface of a sphere, any of those infinite planes will cut the sphere into two parts, and the shape of the intersection is always a circle. Think about marking two points on an orange and making a cut. You have infinitely many choices on how the straight cut works, depending on the angle of the knife to the surface of the orange.

If instead we think of the globe, we know that for any point on that globe there is a unique point that is exactly at "the other side of the world". The two most obvious such points are the North and South poles. For me, sitting here in Oakland, California, the other side of the world is near Madagascar off the coast of southeast Africa. On this transparent globe picture I nicked off the Internets, the point in about the middle of the picture somewhere in the North Atlantic is opposite the island of Tasmania, which is shown in dark bluish gray. The technical term is that the two points are antipodal, and every point has a unique antipode.

If I were in an aircraft with a huge gas tank, and I started flying "straight" in any direction, never changing my heading, I would eventually fly over the antipode of Oakland, and then if I flew even farther would eventually pass over my own house again. The path I would fly is called a great circle, which is to say that on a sphere, while any intersection with a plane creates a circle, the biggest possible circle is the one that cuts the sphere into two equal parts. The equator is obviously such a great circle, but any plane that includes two antipodes creates a great circle. If we think three dimensionally, it also includes the center of the sphere. This is how we measure distance on a sphere, the shortest path from point A to point B. Think of making a cut on the plane that includes point A, point B and the antipode of A, call it A'. If, for example, I wanted to visit Padre Mickey, the shortest flight path from Oakland to Panamá would be the one that if it continued on without changing direction would take me to Madagascar. Likewise if I were to visit New York to see my Internets buddies, the unique shortest path is the one that is on the great circle that includes Oakland, New York and that point near Madagascar we can call the Matty Boy South Pole.

This is the reason that on long trips between points in the Northern Hemisphere, the flight plans are called polar routes, since they fly farther north than seems to makes sense intuitively, though they don't actually go over the North Pole.

Hope this helps.


pissed off patricia said...

Okay, I followed you a lot farther than I usually can. I don't know if you are making things simpler or if I'm getting smarter. I have a feeling the credit goes to you

dguzman said...

What PoP said!

ken said...

Yeah, that's a good explanation. Another thing folks need to realize is that the typical flat map really stretches out the polar regions, so the great circle route looks stretched, too. Go get a globe and look at, say, New York & Tokyo. It'll probably be apparent that the short way goes way up over northern Alaska, and not through the mid-Pacific near Hawaii. If you don't see it, go get a piece of string & pull it tight between those points.

- ken

Anonymous said...

This one is by far my second favorite Wednesday Math post. Its appeal is twofold:It's math-ey, and it makes me want to travel. Thanks!

Distributorcap said...

you are my hero......

i knew you would get me to europe on a tank of gas.



John V said...

What if you start in Oakland and fly due east? You never change your heading, but you do not pass over the antipode of Oakland, though you do eventually return to Oakland.

Matty Boy said...

Hey, John. Flying "due east" forces us to use the rudder. It looks like a straight line if you follow it on a planar map, but not on an actual sphere.