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Walter Guinon said-

>George H has the right idea, Bowditch gives his approach as an approximation to
>the shortest distance. Howerver the shortest path, subject to a maximum
>Latitude, is given by a GC thru the departure point that is tangent to the
>limiting parallel, then along the parallel to another GC thru the destination
>point and which is also tangent to the limiting parallel.
>
>When this approach is studied on a gnomic chart the extension to arbitraty
>curves which exclude certain areas (e.g. land) seems clear.
>

Response from George Huxtable.

Yes, Walter is quite right. I have done a bit more pondering and now accept
that my approach, even when using a globe-and-string, was still
over-simplistic.

In my first contribution on this topic I posed this question- "Similarly,
when reaching X why not then turn through an angle onto a new great circle
between X and Yokohama?"

With hindsight, that could never provide a solution to the shortest-path
problem. It can never involve a sudden turn through an angle on to a new
course, because if one's path involves such a change of course, one can
always shorten it a bit further by cutting across, or smoothly rounding,
the angle. So it's intuitively obvious that the shortest ship's-path must
be a smooth curve with no corners, a condition that's met by choosing an
initial great circle (and a final one) that merges smoothly at a tangent
into the constant-latitude leg of the course. Just as Walter (and Bill
Noyce too) have suggested.

In my second mailing, I said-

"I had indeed missed something, by ignoring the possibility that under some
(but by no means all) circumstances the great-circle path between X and the
end-points could take the vessel nearer the pole than X is. Thanks to the
others for pointing it out."

I suspect that statement too may be inaccurate. Perhaps instead of "under
some (but by no means all) circumstances", I should have said "in all
circumstances". I'm still pondering about that.

The problem is certainly a bit knottier than I first thought. My thanks to
the other contributors who have understood it better.

We all seem to agree (including Dan) that the procedure quoted by Dan Allen
from a website is certainly NOT the best way to do the job.

In my 1981 edition of Bowditch vol2, the relevant formulae are listed in
art. 1016, "Great-circle sailing by computation, on page 604.

George Huxtable.



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george@huxtable.u-net.com
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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