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To George and others that are still interested in the origin of this thread:

In a recently published history of navigation (in French - alas George!),
 the
author treats "Non-graphical methods of position fixing" (p. 374 et sequ.).
One of them is the method proposed by Philippe Bourbon in his "La navigation
astronomique" (Paris 2000), which received a price of the "Acad?mie de
marine" in 2001. I have not tried to follow the math involved (it uses
matrices and vectors), but if anyone is interested I could make a scan and
send it off-list.

Regards, Wolfgang

-----Urspr?ngliche Nachricht-----
Von: Navigation Mailing List
[mailto:NAVIGATION-L@LISTSERV.WEBKAHUNA.COM]Im Auftrag von George
Huxtable
Gesendet: Donnerstag, 8. Juni 2006 22:14
An: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Betreff: Re: Position from crossing two circles : was [NAV-L] Reality
check


When I wrote-

| The method has also been described in
| "The K-Z position solution for the double sight", in European
Journal
| of Navigation, vol.1 no, 3, December 2003, pages 43-49, but that
| article was bedevilled by printing errors that render it
more-or-less
| unintelligible, which were corrected in a later issue. Not to
mention
| several serious errors and misunderstandings by the author, which
have
| never been acknowldged or corrected in that journal.

I failed to give that author's name, which was, as Herbert Prinz
states below, Herman Zevering. Zevering has shown many errors and
misunderstandings, later in that article and at great length in
Navigator's Newsletter, and has even recently had an erroneous
publication in the prestigious Journal of Navigation (of the RIN,
London). However, I think that his treatment of two intersecting
circles from a stationary observer (if nothing else) is sound, and
I've used it as the basis of my program, which appears to give correct
results whenever tested (though not very thoroughly).

When an observer sees a body at a certain geographical position with a
certain altitude, there's no doubt that he is then on a circle of
radius equal to the zenith angle, centred on that GP. The problem
arises when having made that observation, he then travels, through a
certain course and distance, before making another observation. The
locus of an observer who was somewhere unknown on that circle becomes,
after that displacement, not a circle at all. It's distorted, and the
greatest distortion occurs in directions at 45 degrees (and 135, 225,
315 degrees) to the direction of travel. Zevering has assumed the new
locus to be the original circle with its centre displaced by the same
amount that the observer has travelled, but that is wrong. However,
his next step computes the mathematical intersection of two circles,
so if one of these is not a circle, the answers will be wrong. And
they are.

That's where Micheal Dorl has gone wrong, in writing-

"Since one has a good idea of the course and speed between any two
observations,
it seems to me the problem is to determine points on the equal
altitude
circles separated by that vector."

The trouble is, that they are not both circles.

In a later message, Herbert Prinz wrote-

"Right. And this has to be done point for point along the circle, not
by
just moving its center by that vector. The "circle" isn't a circle on
a
mercator projection to start with and changes its shape when being
moved
north or south. Said vector translates to different delta longitude at
different latitudes, so by how much do you shift the GP of the star?
You
know this only after you have the latitude of your fix."

Herbert has it right, but it's NOTHING to do with Mercator projection.
The problem is that when transferred in that way a circle, even when
drawn on the spherical Earth, or on a model globe, is no longer a
circle. It can't be drawn with compasses. It becomes a sort-of
egg-shape, with a blunter end and a sharper end, though not in any way
"elliptical."

A position circle, displaced by shifting every point on its periphery
through the same course and distance, remains a circle only if it is
small compared with the size of the Earth: that was the special
situation that Mixter was addressing, of near-zenith sights, in the
passage that Mike Burkes brought to our attention.

No doubt, by a simple iteration process, knowing the general
whereabouts of the intersection point, one could readily readjust the
displaced circle to arrive at a precise final value. But that's taking
us away from the basic "intersection of two circles" procedure.

Herbert Prinz wrote-

| Zevering's article is comically absurd. He tries to solve the
combined
| altitude problem for a running fix by advancing the first circle of
| equal altitude. His method for doing so is seriously flawed, leading
to
| wrong results. Noting the discrepancy with a solution obtained by
the
| well founded intercept method, Zevering concludes that the latter is
| unreliable!
|
| Actually, there is no rigorous method of advancing the circle of
equal
| altitude, but an approximate solution can still be had. How this is
to
| be done properly was shown by A'Hearn and Rossano in Navigation,
Journal
| of the Institute of Navigation, Spring 1977, Vol. 24. For the simple
| case where both altitudes are taken simultaneously, see S. Howell,
| Practical Celestial Navigation. Other variants of solutions to the
| combined altitude problem (by Kotlaric, Dozier, etc.) are described
in
| editions of Bowditch of the late 60s to 80s.

I haven't come across the A'Hearn and Rossano paper, but it certainly
seems worth looking up. I've had a look at the descrptions of the
Dozier and Kotlaric methods, in my 1977 Bowditch, but they don't seem
to be very relevant to the present discussion. If I am
misunderstanding, perhaps Herbert will explain.

As for Andres Ruiz' program, it's of an era that has left me way
behind, and I have no hope of following what it does. A few words of
explanation might help. Where do the numerical values at the end, two
pairs of GHAs, Decs, and altitudes, come in?

Thanks to Andres for the reference he has indirectly given, which
seems highly relevant; it's "Advancing celestial circles of position",
Metcalf T R, Navigation vol 41 no.2, Summer 1994, pp207-214.

George.

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

   

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