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From: George Huxtable (no email)
Date: Thu Jun 15 2006 - 18:30:51 EDT
As these things can do, the discussion has drifted to a somewhat
different topic. It was about intersecting circles, corresponding to
two bearings taken by an observer from a single position. Now it's
about what happens to a position circle as an observer's locus whan
the observer moves his position. To mark that occasion, I have tried
changing the threadname accordingly.
For me, this discussion has come at an interesting moment. I am
expecting to publish, in the next issue of "Journal of Navigation"
(the London publication, of the RIN, not the US version), a refutation
of a proposal (by K H Zevering) to compute positions by intersecting
circles. That proposal presumes that a circle (as a locus of an
observer's position) stays a circle when the observer shifts across
the surface of the Earth through a known direction and distance. In
that respect, my colours are now nailed to the mast, so if it turns
out that I am wrong (and a circle, shifted in that way remains a
circle), then I will look particularly daft. So I am interested in the
views of Nav-l members on such matters, being perhaps the most
knowledgeable community that's around. If I am wrong, I would like to
know about it, as soon as possible.
Responding to a posting by Micheal Dorl, as follows-
| "Assume a equal altitude circle at a
| few hundred miles in diameter that just falls short of the pole by
say one
| mile. Now advance that circle by 6.28 miles west. The circle
grows a kind
| of upside down J shaped figure with the hook of the J surrounding
(but not
| enclosing) the pole. "
Frank has written
| Which tells us that there is something terribly wrong.
As I see it, Michael is absolutely correct, and the only thing that's
wrong is the expectation that when you shift every point on that
original circle west by 6.28 miles, it will stay a circle. What
Michael has done is to consider an extreme case, which is often an
illuminating way to argue.
Frank hasn't actually stated whether or not he considers that the
locus of a circle, advanced (at every point) by a certain course and
distance, must remain a circle, and I ask him to ponder the matter
carefully, draw the picture and put in some numbers, and tell us just
what he thinks is "terribly wrong" about that extreme picture Michael
Dorl has presented.
In a later message Herbert Prinz wrote-
"I moved a penny across a celestial globe to visualize
Zevering's procedure and forgot for a moment that this was exactly
what
I had argued against earlier!"
And Frank has replied, in a way that implies scepticism-
"So I'm up in the high Arctic in the winter driving across the ice. I
get out
my bubble sextant and through broken clouds I get an altitude of
Polaris at
87 degrees. I see Jupiter through a break in the clouds a minute
later low in
the sky but I don't have time for an altitude. Now I travel for sixty
miles
in a straight line 30 degrees to the left of the azimuth of Jupiter
at the
time of the first sight (it's not possible for me to say "I traveled
'north'
for sixty miles" since I can't measure that, right?). The clouds
break, and I
then shoot an altitude of Jupiter and find it's [pick a number]
degrees high.
I want to advance my initial circle of position for Polaris and cross
it with
the position line for Jupiter. How do I do that? Is my advanced
circle of
position distorted?"
Well, Frank has chosen an awkward example. The only circles that
remain unchanged as circles, when their positions are advanced in that
way, are circles that are small compared with the size of the Earth;
circles, that is, that correspond to an altitude measured near to the
zenith. That is exactly the case he has chosen for his example, taking
an altitude of 87 degrees. That's a position circle only 3 degrees, or
180 miles, in radius, which is tiny compared with the Earth, and plane
geometry is a pretty good approximation in that case. And of course in
plane geometry it's perfectly true that when you shift every point on
the surface of a circle through the same distance and angle, you end
up as circle with a displaced centre. It's just circles on a sphere,
of a finite size compared to the sphere's diameter, that suffer
significant distortion.
Even so, I think he will find that when he moves even a small circle,
and takes it really close to the singularity at the pole, it will
behave just as oddly as in Michael Dorl's example.
But Frank's example diverges from the situation we are considering in
another way. Depending on where, on the initial locus, the observer
happens to be, the azimuth of Jupiter (as well as its altitude) will
differ. Whereas we are considering a situation when an observer, no
matter where he is on the initial position circle, then travels with
the same course and direction. That's the question we have set
ourselves, and are doing our best to answer. I suggest, then, that
Frank's example is something of a distraction from that aim; perhaps
he will reformulate it..
But it's necessary to keep aware of exactly what is being considered.
It ISN'T a case of sliding a penny, or a wineglass rim, around on the
surface of a globe. In that situation, then the circular rim remains a
circular rim, and by definition, there's no distortion of it. But in
that situation, it will be found that all points on that rim are NOT
all moving through the same course and distance. We are looking at the
converse, when every point on the locus DOES shift through the same
course and distance, and hence the locus does not remain circular.
I hope that anyone who is still unconvinced by those arguments and
Michael Dorl's example, and believes that a circle on the globe,
advanced in the way we are considering, remains a circle, will stand
up to be counted, and we can thrash the matter out further (and
perhaps enjoy it).
George.
contact George Huxtable at
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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