NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Advancing a position circle. was : [NAV-L] Position from crossing two circles
From: George Huxtable
Date: 2006 Jun 15, 23:30 +0100
From: George Huxtable
Date: 2006 Jun 15, 23:30 +0100
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 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.