Rhumb Lines and Great Circle Routes
Date: 09/24/98 at 14:49:50 From: Ira Leeds Subject: Rhumb Lines and Great Circle Routes In a recent Geometry class, we came across a question about the shortest distance between two points. The answer the text gave was three separate solutions. a) a straight line b) a rhumb line and c) a great circle. We determined that there was a different answer for each type of geometry. We were wondering, what is a rhumb line and what is a great circle and how do they apply to shortest distance between two points?
Date: 09/24/98 at 17:52:00 From: Doctor Rick Subject: Re: Rhumb Lines and Great Circle Routes Hi, Ira. Recently I helped a Coast Guard lieutenant work out distances along rhumb lines and great circles. Straight lines would be so much easier - but hard to travel! I'm sure you talked about the fact that, although the shortest line between New York and London is a straight line, you'd have to tunnel far below the surface of the earth to travel along it. We're really interested in the shortest distance on the surface of the earth between two points. You might think that if you take a map of the world and draw a straight line from New York to London, that would be the shortest distance between the two cities. But it depends a lot on the type of map - that is, the map projection. You can probably find at least two kinds of world map in your school, a Mercator map (that's the one with Greenland as big as all of South America - see the following site for more information: Mercator's Projection: Robert Israel, Univ. of British Columbia http://www.math.ubc.ca/~israel/m103/mercator/mercator.html and an equal-area map (there are different kinds, but they are popular because they don't shortchange the tropical countries the way the Mercator does). If you lay a ruler between New York and London on each map, you will find that the two lines pass through different places en route. They can't both be the shortest route - and in fact neither of them is. The line you made on the Mercator map is a rhumb line. After what I just said, you might think, "so what?" But the Mercator map does have a special property that other maps don't: you can read off a compass reading from that line you drew. If you got in an airplane in New York and headed in that direction, you would eventually get to London. (I'm ignoring the problem that compasses don't point to the north geographic pole, but to the north magnetic pole. That's another subject.) If you did the same with an equal-area map, you'd miss London. I already said that a rhumb line isn't the shortest distance between New York and London. It's just the easiest route to follow. Look again at that rhumb line you made on the Mercator map. Then think about the fact that airplanes that fly between New York and London routinely fly over Newfoundland. That's far to the north of the rhumb line. Why do they fly there? Because the actual shortest distance between two points on the surface of the earth goes that way. Now take a globe. Find New York and London, and turn the globe so that they are on a line around the middle of the globe, as if you moved them to the equator. Where is Newfoundland? It should be on the "equator" too. That line around the middle of the globe is a great circle. One way to think about it is that if you sliced the globe right down the middle in any direction, along a plane that passes through the center of the globe, the line you made on the surface would be a great circle. On the cross-section of the globe that you exposed, you could draw the true shortest distance between New York and London, a straight line that cuts across the curve of the earth. But the arc of a great circle is the closest you can come to that line without going beneath the surface. So there you have it. In Euclidean space, a straight line is the shortest distance between two points. But in a sense, we don't live on a Euclidean world. If you want to go beyond the horizon, a great circle is the shortest distance between two points. A rhumb line is not the shortest distance in any sense except on a Mercator map, but it is very useful. In fact, those airplanes probably don't really travel a great circle; they approximate it with a sequence of rhumb lines so they only have to change compass bearing once in a while. (Well, with computer navigation, maybe they do travel true great circle routes now.) I hope this answers your question, but I hope it doesn't answer everything, because there is a lot to explore in the topic of maps and navigation. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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