Wednesday, November 26, 2003

Great circles!

Ok, what's a great circle? It's a concept from spherical trigonometry, and...hey, wait, where are you going?

It's not that big a deal, really. All you need are a sphere and a plane. Now make sure that plane goes through the center of the sphere. The circle described by the intersection of the sphere and the plane is a great circle.

Depending on how you orient the plane and sphere, there are an infinite number of great circles. Pick any two points on the surface that aren't at opposite poles and there's exactly one great circle that contains them both. Parallels of latitude are not great circles. Meridians of longitude are all great circles, and but for political perversities time zone boundaries would be too.

So who cares? Navigators, for one. It turns out that the shortest difference between two points on the surface of a sphere along the surface of the sphere follows a great circle - the one that contains your start and endpoints. What do you know? - that happens to describe driving, or flying at a constant altitude, or sailing.

Soon I'll be making a round trip between DC and central IL by car. Looking at the atlas, it looks like the shortest way to go is by way of I-70 to Indy, then into the wilderness.

Then I decided to see what Yahoo said. It pointed me along I-80, up near Cleveland, Toledo, South Bend, the south side of Chicago. It sure looks farther. What the heck, let's see what AAA says. Yep, same thing.

Alright, I'll trick it. I'll route a trip from DC to Indy, and from Indy to the sticks to force it onto I-70. It worked, but the total mileage was higher. Sheesh, just look at the map - how can that be?

The problem is that I have a flat map of a spherical object, which causes distortions and in particular does not show great circles. One result is that east-west distances appear greater than they are. And it gets worse the farther north you go, until you reach the North Pole and every direction is south. Over a distance of several hundred miles the effect is significant. So the I-80 route only looks like a longer path because it is further north and thus is more distorted by the map.

If my map had shown great circles, they would lie above and to the outside of straight lines drawn between the endpoints, coinciding only when you're going straight north or south. If you were traveling exclusively on latitude and longitude lines and you wanted the shortest path between two points, you would want to go east or west before you went south, or after you went north. Of course if you're in the southern hemisphere the rules are different, and I'll leave that as an exercise.

Now if I really wanted to open a can of worms I could get into why roads are routed the way they are. But as much as I wish this were dictated by engineering considerations, the reality is that there is all manner of political skulduggery behind it. One example, from Illinois, might wind up in a future post.

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