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Re: Advancing a position circle.
From: George Huxtable
Date: 2006 Jun 17, 18:03 +0100
From: George Huxtable
Date: 2006 Jun 17, 18:03 +0100
In "George's example, revisited", Alex was right to question my claim that it would be impossible to draw a circle on the globe that passed through those three points. As he said, you can always draw a circle that passes through any three points, on a sphere or on a plane. What I was presuming, without stating it or even thinking about it, was that to preserve the symmetry of the picture, the new centre must be somewhere along the specified direction of motion; that is, due North from 0N, 0W. And in that case, there's no position on that line that's equidistant between the three specified points. Alex and I now seem to agree about all aspects of what happens when you shift every point on a position circle, except for one minor matter, from "George's example, corrected", in which he states-. | It seems that both you and I made a trivial mistake | in the calculation in this example (cited below). | The circle of the best fit is centered at | the point (1,0), but its radius is not 60 degrees | but 59d 53'. | This gives you approx. 7 miles distance, | between this circle and the shifted position line. | | So the answer is closer to 7 miles rather than 14 miles. | | Of course this does not change the main conclusion | that the deviation is not negligible. | Just wanted to correct the number for the case that | you might wish to include this example to your paper. ======================= I suggest that the radius "circle of best fit" that Alex refers to, which comes into the matter simply in order to establish the deviations from it, remains at 60 degrees, not 59d 53', when you take into account the whole advanced "circle". The radius of that locus, at an azimuth of about 315 deg from the origin, at (46N, 45W), that I considered, is indeed shrunk by 14 miles, as we have agreed. And so it is at the symmetrical azimuth on the other side, at an azimuth of about 45 deg, at (46N, 45E). But consider the corresponding azimuths to the SE , at 135 deg (44S, 45E) , and SW, at 225 deg (44S, 45W). In those cases the radius is ENHANCED, by about the same amount, 14 miles, so taken all round, any "circle of best fit" will still have a mean radius of about 60 degrees. But anyway, the deviation that I am pointing out is not with any "circle of best fit", but with a circle created on the naive and erroneous basis that it should have the original radius but its centre displaced by the course and distance; in this case, 60 deg radius with its centre shifted to (1N, 0W). That was the published proposal that I have been trying to correct. I have described the new locus as a sort-of egg-shape; and so it is, in some respects but not in others. Here follows a short discourse about the outline of an egg, as laid by a hen, as seen fron one side rather than from one end. The egg outline departs from a circle in two ways. First, it has a long axis and a shorter one; like the outline of a rugby ball rather than a football. Second, it has a "blunt" end and a "sharp" end. The locus I am discussing departs from a circle in that second respect, but not in the first, so it is only partly eggy, not entirely so. Peter Fogg is worried about the fundamental reason behind all this. First, I should make it clear that it's nothing whatever to do with the fact that the Earth deviates slightly from a sphere. Everything has assumed an exactly spherical Earth. I can suggest an illustration that may help, by showing the converse of my argument.. No doubt he owns some sort of globe of the Earth. Take something with a circular rim, like a wineglass, which has a radius that's a large fraction (half or more, say) of the radius of the globe. Put four markings at 90-degree spacings around the rim of the glass, marking two opposites to be North and South. And put four more markings in between, at the 45 deg positions. They don't need to be specially accurately done. Now put the rim over some position on the Earth, such as (0N, 0W), with the N and S marks on the 0 deg meridian and two others on the Equator. Next, slide the glass due North by a few degrees, keeping the N and S marks on the 0 deg meridian. First, check what happens to the marks at E and W. They also move due North; no problem there, though that motion will actually be a bit less than the N and S marks. But look hard at the in-between markings, at the four 45-deg positions. They are shifting, not along a North-South meridian, but at a completely different angle. Their courses differ considerably. And there you have it. To keep a true circular shape, when you shift a circle, different parts of the circle must move in different courses, and by differing amounts. Conversely, if you insist that they all must move along the same course by the same amount, the result cannot be a circle. Not a formal proof, I know, but is it sufficient to convince Peter Fogg? 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.