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"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.

   

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