# NavList:

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

**Re: Still on LOP's**

**From:**Rodney Myrvaagnes

**Date:**2002 Apr 20, 01:40 -0500

On Fri, 19 Apr 2002 16:39:46 -0400, Noyce, Bill wrote: >I've also been thinking about George's assertion that the >cocked hat contains the tru position just 25% of the time, >and, much to my surprise, I've convinced myself it's true. > a counterexample that covers practically all plausible cases: Taking George's simplifying assumption of a rectangular probability distribution, let's be sure we know what that means. Any probability distribution must sum to 1 over all possible cases. Or, the integral from minus infinity to plus infinity of [P(x) dx] =1. Hence, the probability of the obsevation falling in any intervalof x within the non=zero range is exactly the fraction of that range the interval takes up. With three lines of position, all independently measured by the same method, we must assign the observations the same probability distribution. One possibility is that they are just wide enough to make all three observations possible. That results in the traditronal center point of the triangle being the only non-zero probability. Another possibility is to make the rectangles just cover all the other observations. If the cocked hat happens to be equilateral, the non-zero area expands to fill the triangle, and is zero everywhere outside. As the triangle changes away from equilateral, a band of non-zero appears outside the two long sides, but the inside of the triangle remains as strong. Eventually the two ouside triangles will equal and pass the area inside. To treat the result that way is to ignore the geometric reason the band got wider--that two of the LOPs are nearly parallel, and the distance of their crossing from the third LOP does not detract from the third LOP. A reasonable band to apply in this case would be the width that allowed the nearly-parallel LOPs to reach the two closely-spaced corners on the third (crossing) LOP. Rodney Myrvaagnes J36 Gjo/a "Curse thee, thou quadrant. No longer will I guide my earthly way by thee." Capt. Ahab