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Dear George,
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.

Alex

On Sat, 17 Jun 2006, Alexandre E Eremenko wrote:

>
> George's example of a "shifted circle" was the following.
> Begin with the circle centered at
> (Lat, Long)=(0,0)
> of radius 60 degrees.
> Shift every point 1 degree N.
> The resulting figure is not a circle,
> and the best fitting circle deviates by more than 14 miles from it.
>
> All this is correct.
> Indeed, choose 4 points on the original circle as follows:
> A=(0,60), B=(0,-60), C=(60,0) and D=(-60,0).
> They are shifted to the new points
> a=(1,60), b=(1,-60), c=(61,0) and d=(-59,0).
> Now consider the circle best fitting these four points.
> It is clear from symmetry that the center of this best
> fitting circle should have longitude 0.
> The equidistant point from c and d is x=(1,0).
> However, the distance from this point x to a (or to b)
> is about 50d46' (of a great circle). Simple trig is used to
> compute this.
> So the deviation of our figure from any circle is about 14
> miles, as George stated.
>
> This shows:
> a) I was wrong when I wrote that the "size of the circle
> is irrelevant, but only relevant is its distance from
> the poles". Size of the circle is also relevant.
> b) It is a bad idea to use position circles of large radius
> for running fixes. In any case, it is wrong to think that
> a shift of a position circle will be close to a circle.
> c) The only case when the use of position circle
> (as opposite to a position LINE on a Mercator chart) is
> justified is when the circle is small.
>
> Thus all statements of George on the subject (except that one cannot draw
> a circle through 3 points) were correct.
>
> Alex.
>

   

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