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/