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From: George Huxtable (no email)
Date: Sat Jun 17 2006 - 13:03:55 EDT
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
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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