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Subject: Intermediate points in great circle courses
From: Lu Abel (lu.abel@XXX.XXX)
Date: Sun Jul 25 1999 - 14:33:26 EDT
A friend of mine who is not a sailor but an avid electronic gadgeteer
showed up the other day with his new handheld GPS. He asked me how far
away the destination point could be in finding course and distance. "As
far as you want." To demonstrate, I pulled out an atlas and we punched in
the L/Lo of Times Square in New York City (we're in San Jose, California).
He noticed the indicated course was a bit more northerly than one would get
from the atlas. I told him the GPS was calculating great circle courses,
which always run more pole-ward than straight-line (rhumb line) courses.
He's a smart guy -- he's heard of great circle courses and understands what
they are.
He asked how to calculate a great circle course. I pulled out the USPS JN
text and looked up the spherical trig formulae. He whipped out his laptop
computer and in a couple of minutes had an Excel spreadsheet which took two
L/Lo's and calculated distance and initial heading. The results were in
excellent agreement with what the GPS was displaying for course and
distance to NYC.
A couple of days later he asked: "Lu, how do you steer a great circle
course? How do you know what your headings should be as you sail or fly
along a great circle?"
I told him that one would plot several intermediate points of the course on
a Mercator chart and then steer the rhumb-line courses between them. Then
I realized I didn't know how to find intermediate points.
My copy of Bowditch has grown feet and disappeared. Dutton's only advice
(as well as the USPS's) is "use a great circle chart and transfer points to
your Mercator chart."
Since a great circle course is one side of a spherical triangle with the
start and end points and the elevated pole as vertices, I would assume I
could calculate the latitude of a point on the course for any DLo, but my
spherical trig is really rusty and I don't have a book of formulae.
I'll also point out that one could solve the problem in a fraction of a
second on a computer using a technique known as iteration: If I want the
latitude of a point on the course for some intermediate longitude, guess
one (say by simple plane trig using the initial course), calculate the
course to it from the starting point, if the course is more pole-ward than
the great circle then the assumed point's latitude is too great and it
needs to be reduced; if the course is more equator-ward, then the latitude
is too low. The key is to adjust the latitude by an amount which is
proportionate to the course error. If one does this repeatedly, the
assumed point will get as close as one wants to the great circle.
Thanks for helping fill out what I suddenly realize is a hole in my knowledge!
Lu Abel
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