# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Cocked hats, again.
From: Paul Hirose
Date: 2007 Mar 15, 23:44 -0700

```My cocked hat "thought experiment" confirms Huxtable's Theorem.

Imagine three perfect observations, so the LOPs all meet at a point.
That's the true position of the vessel.

In the real world all observations have error. None of the LOPs will
pass precisely through that point. To simulate observational error, move
each LOP some random distance toward or away from its body.

Moving the first LOP generates a triangle with the true position at one
vertex. For the remaining two LOPs (which meet at the vertex) there are
four possible permutations of toward and away. Only one yields a
triangle containing the true position: 1) move one LOP in the direction
that enlarges the triangle, 2) move the other LOP in the direction that
enlarges the triangle.

If you move the LOP in step 1 the other way, the true position is
outside the triangle, and on the LOP that hasn't moved yet. Then no
matter which direction you move that LOP, it's impossible to form a
triangle containing the point.

None of this depends on how the sights are distributed in azimuth. As
long as all three are equally likely to err toward or away, there's a
25% probability that the true position lies within the triangle.

Skewing the toward/away probability will change that. For example, if
the azimuths are about equally spaced and you forget to apply the dip
correction, it becomes *more* likely that the cocked hat contains the
true position!

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