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Re: spherical geometry

At 8:56 AM -0500 8/19/04, RAUBER, JOEL wrote:

As a side note, the airlines and mariners have long been interested in the
solution to this problem. And certain map projections have been designed to
facilitate the solution.

If memory of my days doing navigation while in the Navy serves, There
is not a single heading one can take to follow a great circle route
from point A to point B, since the great circle route crosses each
meridian at a different angle, and it is that angle that determines
the compass heading one takes at that point. The straight line path
on a Mercator projection map (called a "rhumb line") does have a
constant heading, but it is not the great circle path.

Given that, I don't think that ships or aircraft would be interested
in a single heading to follow for an extended trip--one for which a
great circle path is significantly different from the rhumb-line
path. As a practical matter, on a trans-oceanic trip, a ship's
navigator will calculate the great circle path (and I have forgotten
the method used to do that on a Mercator projection), and then
approximate that path with a series of rhumb-line segments, so that
the ship's heading will change periodically during the trip. I recall
vaguely that the rhumb-line segments typically were about 5 degrees
of longitude, each, or some 300 nautical miles or less--roughly one
day's steaming.

Things may have changed dramatically with the advent of GPS
navigation, which occurred long after I got out of the navigation
business. I would expect that computerized nagivation systems will
automatically calculate the heading on a great circle route, leaving
the navigator with precious little to do, once a chip is clear of
harbor restrictions.


Hugh Haskell

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