A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Jim Hickey
Date: 2006 Dec 15, 16:39 EST
The comment made in an earlier posting that there is a 1 in 4 chance of your position being inside the cocked hat prompted me to having a closer look at the potential geometry involved.
I understand the 8 possible triangles based on a particular LOP being away or towards. The problem I have is that if I think about an LOP with an intrinsic random plus or minus error (let’s leave systematic error alone for now) it seems extreme to consider only the triangles made up of the LOP’s being at one extreme or the other, i.e. maximum away or maximum towards is not reasonable.
If I draw a diagram using 3 pairs of lines each at 60 degrees to each other you quickly see the 6 small triangles that would create very nice cocked hats in a star pattern exactly where you are not. Fair enough. You also see the 2 very large triangles that you are in. What is interesting with the 6 small triangles are that not only are you not where you should be but you are actually outside the error range of 1 of the pairs of lines. When the assumption is made that the LOP is either maximum towards or away, you are looking at the statistically least likely situation. Having at least one LOP closer to the true position is more likely and having any one LOP closer improves the position one would determine,
The attached diagram demonstrates the point and I believe is a good diagram for someone who is studying CN to contemplate. It shows 3 ideal LOP’s with their associated away and towards error limits. Granted it takes advantage of symmetry to demonstrate the point but you can see that there are 6 possibilities to create a cocked hat that does not contain your position and there are 14 that you include or have a point on the edge of the cocked hat. Considerably rosier situation than a 1 in 4 chance. Another interesting observation is the series of 6 points that are described by 3 LOP lines running through them at exactly not where you are!
So as has been mentioned in recent postings, cocked hats don't necessarily contain your true position and small cocked hats (or exactly crossing lines for that matter) should not imply greater accuracy but, statistically, the position that you define with the cocked hat is likely to be better than the 1 in 4 argument might suggest.
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