# NavList:

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

**Re: Still on LOP's**

**From:**Geoffrey Kolbe

**Date:**2002 Apr 21, 11:30 +0100

"A picture is worth a thousand words" so here is a picture: http://www.pisces-press.com/graphics/hats.jpg There are eight small drawings in hats.jpg. Each drawing shows three known geographical positions, which could be prominent landmarks, represented as black spots. The correct bearings to your actual position are shown in black. The bearings all meet at the actual position. This is a zero error, theoretical construct. But if you have a map which shows your actual position and the three land marks, It is possible to draw such a diagram on the map. If bearings on the landmarks are taken from the actual position, there will be an error in the bearings so measured. 50% of the time the error will be to the left, and 50% of the time the error will be to the right of the landmark. Let us call this L and R errors. The drawings also show simulated real life bearings taken on the landmarks and plotted in red. The bearings have either a left or a right error from each landmark, indicated by the letter at each landmark. There are three landmarks, the bearing on each can have a left or right error, so there are eight possible combinations of errors - and so eight drawings showing these combinations. These errors result in a cocked hat where the bearings intersect. You will note that only in two cases do the resulting cocked hats enclose the actual position. A little thought will also reveal that for any given arrangement of landmarks, two combinations of error will ALWAYS enclose the actual position, and six combinations of error will NEVER (and can never) enclose the actual position. Since each of these scenarios is equally likely, the chances are one in four that the cocked hat due to the intersection of the bearings taken from three land marks will include your actual position. I have omitted a few considerations due to symmetry, but I am pretty sure that this approach constitutes the basis of a formal proof of the assertion that the cocked hat resulting from the observation of three objects will only enclose your position 25% of the time. Not 23%, or 26%, or 17.5%, but 25%. A similar argument can be put forward for LOP's, where the cocked hat is due to errors in measuring altitude. Geoffrey Kolbe.