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Subject: Re: On LOPs
From: Steven Tripp (tripp@XXX.XXX)
Date: Mon Apr 15 2002 - 23:06:04 EDT
On 4/16/02 9:36 AM, "Jared Sherman" <jared.sherman@XXX.XXX> wrote:
> So far I haven't seen any justification for the "most probable" position, much
> less for the use of an ellipse (which has 2 focii) rather than a circle (with
> one focii, i.e. center) when using 3 or more LOPs (which would indicate a need
> for 3 or more "focii").
The ellipse is based on formulas from the Nautical Almanac. I wrote the
program more than twelve years ago based upon those formulas, but my oldest
Almanac (1992) doesn't seem to contain them anymore.
The confidence ellipse is just a two-dimensional version of the standard
one-dimensional confidence interval used in statistics.
The DR position is irrelevant and is plotted only for navigational
convenience.
The MPP seems to be "pulled" towards the LOP intersection which is most near
90 degrees. In the ideal situation the three LOPs will cross at 60 degrees
and form an equilateral triangle. The MPP will be at the center of the
ellipse and the ellipse will be a circle.
When the intersecting angles of the LOPs are quite different the ellipse
will be elongated. The MPP will be at the CENTER of the ellipse, not at a
focus. The center will NOT be the center of the cocked hat.
Note that part of the cocked hat may be outside the confidence ellipse. The
size of the ellipse depends on the confidence level. A 99% ellipse will be
large. A 50% ellipse will be small.
The number of LOPs does not matter, except that with more lines the size of
the ellipse will be less, indicating greater confidence. This is a standard
statistical effect. A larger (random) sample reduces the probability of
error. Each LOP you observe is a random sample from the infinity of
(randomly different) lines which could have been observed at that moment.
I attach two GIFs: one with three LOPs at quite different angles, and one
with twelve LOPS.
Steve Tripp
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