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From: George Huxtable (no email)
Date: Sun Nov 09 2003 - 08:03:57 EST
Zorbec Legras said-
>By definition a loxodromic track is a logarithmic spiral.
Response from George-
I rather doubt that. It becomes close to a logarithmic spiral near to the
pole, when we can approximate the Earth's surface by a plane, but not
otherwise. Where does Zorbec's "definition" come from?
Zorbec continues-
>By definition a logarithmic spiral never reaches the assymptote point
>which, in this case, the up pole is.
Another "definition". Can we have a reference, please?
We have to be careful here. It depends on the meanings of the words we are
using. What does "never" mean in this context?
If we are following a decreasing logarithmic spiral down to its end-point
at the pole at a constant rate of change of longitude (each turn around the
pole taking the same time), then never, ever, will we reach that end-point,
because an infinite number of turns is required, and there is not enough
time available in the Universe for that to be completed. In that case,
Zorbec's "never" would be justified.
On the other hand, the original problem specified, not a constant rate of
change of angle, but a constant velocity along the ever-decreasing spirals.
In that case, each decreasing spiral takes a shorter and shorter time. And
because the integrated path length along a (decreasing) logarithmic spiral
is FINITE, then the end-point will be reached in a finite time, but after
an infinite number of turns.
=========================
Zorbec continued-
>By the way how many loxodroms are there to join 2 points ?
>Hint : more than 2...
This topic has been aired on this list before, under the threadname "The
pedant's rhumb-line", commencing 10 Oct 2002.
The answer is- As long as the latitudes of those points differ, as many as
you like, depending on how many times you are prepared to circumnavigate
the Earth, Eastward and Westward, along the way.
George.
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contact George Huxtable by email at , by phone at
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