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Big thank you to George and Geoffrey.
Geroge you are right I was taught that the position was located in the
middle of a three point fix. After reading your responce I highlighted my
position lines with a orange highlighter and BINGO I could see the
convergance of the lines. Thank you.
According to the book they say the answer is 40deg 01'N 153deg09'W, I'm sure
they are using a computer for their answer. I agree with Geoffery's answer
40deg 17' N 153 deg 13' W.
Thank you gentleman,
----- Original Message -----
From: "George Huxtable" 
To: 
Sent: Saturday, December 09, 2006 8:52 AM
Subject: [NavList 1850] Position lines, crossing.


>
> Gary Schwartz asked In Navlist 1848, "sorry this time I attached the
> file and sent it to the correct list", a threadname which I have now
> renamed.
>
> "...there are six objects. Maybe plotting 6 objects is too many.  This
> plot is exercise 3-2 from the book 100 problems in celestial
> navigation.
>
> My fundamential question is which sights form the enclosure of my
> position? I'm thinking Alpheratz, Venus, and Rasalhague, however I
> have no basis as to why these and not any others."
>
> And he provided a picture of the plot as an attachment.
>
> =========================
>
> This is a matter that crops up from time and shows up much
> misunderstanding among even experienced navigators, textbook authors,
> and tutors. So it's fine that Gary raises it again and provides an
> excuse to give it another going over. However, some old hands will
> have heard it all before.
>
> The short answer to Gary's question is that NONE of these lines forms
> the "enclosure" of his position. All that can be said about his
> position is that it is somewhere in the vicinity of where the lines
> cross, a patch covered by a broad thumbprint. For the sake of putting
> a dot on his chart, he might take it to be, say, 40deg 12'N, 153deg
> 15'W, but it doesn't matter much exactly where, within a few miles.
> What is really important is that he is aware that it's only a rough
> guide to where he actually is; to within 7 miles or so in any
> direction, by the look of it. And to recognise that it's quite likely
> that his true position may be completely outside the area bounded by
> any combination of those lines.
>
> He wondered if too many objects were being plotted. Not at all. The
> more objects plotted, and the more crossing-lines shown,  the better
> he will be able to estimate the centre their crossings congregate
> about, and the scatter of those position lines around it, which
> provides some notion of how precise the observations actually are.
>
> There are indeed computer programs which attempt to make a
> "least-squares" statistical analysis of such a round of sights, to
> provide a nominal centre-position and an "error-ellipse" surrounding
> it. That can avoid the need for the graphical construction (the
> program can do that for you) but in my view it will gain you little
> over a commonsense view of a plot such as Gary provided; and it can
> sometimes actually mislead.
>
> The simplest situation to consider is that of two such position lines,
> which cross at a point, and that point is what you plot as your best
> estimate of position. But every navigator should be aware that no
> observation is perfect, and that his position lines have an error-band
> which widens them to a few miles across, depending on the
> circumstances of the time, which only he knows best. Things such as
> the size of his boat, the roughness of the sea, the sharpness of the
> horizon line, all give rise to scatter in his result. These are
> matters that the computer doesn't know about, and can only guess at
> from the discordance between many observations. With just two, it has
> nothing at all to go on. It will give the crossing-point, nothing
> else, but the navigator, estimating roughly his confidence in each, is
> in a much stronger position, and can sketch in a rough error-zone
> around his crossing-point, which also depends on the angle of the
> crossing.
>
> A common situation is whan a third observation is taken, to give a bit
> of extra confidence. The three resulting position lines cross to
> create a "cocked hat" error triangle, and it is in discussing this, in
> the past, that so much heat has been created. This is because
> erroneous notions have been so strongly ingrained, as a resut of
> faulty teaching. It has often been taught in navigation classes, and
> probably still is to this day, that such a triangle embraces the
> possible position of the vessel, and that to be safe, a mariner has to
> assume that he is whatever part of that triangle is nearer to a
> danger-point. Nothing could be further from the truth. It's profoundly
> dangerous nonsense; that is not a safe assumption at all.
>
> In fact, if any systematic errors have been properly corrected for,
> and only random scatter remains, the simple truth is this. Only on one
> time in 4 will the vessel be inside that triangle at all, and 3 times
> in 4 it will be somewhere outside it, though in the vicinity. This is
> a simple statistical truth, easily proved, but one that mariners are
> most reluctant to accept, because it is so contrary to what they have
> been taught. Surprisingly, this 1 in 4 rule applies to the most
> skilled observer, just the same as it does to a novice. The difference
> is that the expert's triangles will turn out to be smaller, but still,
> only one in four of those smaller triangles will embrace the true
> position.
>
> Given such a triangle, a least-squares analysis program will do its
> best to assess an error ellipse, based on its size. But you have to
> take such findings with a pinch of salt. Because there's so much
> variation between one such triangle and another, simply as a result of
> random scatter, some will just happen to be tiny in area, just because
> the lines happen to cross closely. When plotting out such a case, the
> observer might think that he had made a particularly precise
> observation, and the computer thinks the same. But an astute observer
> realises that it's just the luck of the draw, whereas the computer has
> no such insight. Only after assessing a run of many similar
> observations can you get a good feel for the overall accuracy being
> obtained in those conditions; not from just a single triangle.
>
> With more observations crossing, such as the six in the example Guy
> Schwarz has given us, a computer has a bit more information to work on
> and can make a better shot at assessing the precision of its resulting
> "fix". And in just the same way, you and I can eyeball those crossings
> of the 6 lines and weigh up the resulting accuracy for ourselves, and
> our intuition will probably arrive at about as useful an answer as the
> computer's. But there's no way that you can draw a boundary-line on
> that diagram and say that the true position must lie within it, which
> is what Guy was asking for.
>
> =======================
> Systematic errors and their effects.
>
> There's a complication. Above, it was assumed that the only errors
> were random ones; that were equally likely to be one way as the other,
> and that any systematic errors had been corrected for beforehand. That
> may not be the case. A careless observer may have got his index
> correction wrong, offsetting all his altitudes by a common amount.
> More insidiously, anomalous dip could be affecting his horizon in an
> unknown way, with a similar result. Various proposals have been made
> for detecting and correcting such errors, but are unlikely to succed
> (in my view) unless those errors happen to be dominant, overwhelming
> the random scatter. It is difficult, often impossible, to unravel and
> separate the effects of such random and systematic errors. But if
> systematic errors are making a significant contribution, they will
> tend to affect that 1 in 4 probability (for triangles) discussed
> above; either increasing or decreasing it, depending on the geometry.
> One reason for making widely-spead observations all round the horizon
> is to average out such systematic errors.
>
> And there can be other type of error, just as systematic as those
> considered above, with a different effect. Such as a clock error,
> which works differently on the altitudes of bodies that are rising,
> compared with those that are falling.
>
> Thanks to Gary for giving me the opportunity to trot out an old
> warhorse and give it a bit of exercise.
>
> 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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