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Dov-
 

I would argue that the problem, as stated, is both solved and trivial.

Given the absence of specific measurements and data, one can only assume the 
gun can be fired freely in all positions throughout a spherical space. Since 
the target, even a target of infinite size, can only intercept the shots 
fired "toward" it, and bullets have a finite limited travel distance, the 
size of the circle containing all bullet holes on the target is limited to 
the smaller of two cases: One being a circle that contains the entire target, 
if the bullets ARE capable of being fired beyond the target's boundaries. The 
other being a circle equal in size to the range of the bullets, i.e. if the 
bullets can reach the target 3 miles from dead center (but fall to ground and 
stop traveling at + miles) then the size of the circle is limited to the area 
the bullets can reach.

The only question left is the mathematically trivial one of whether the target 
chosen will be smaller than the range the bullets can reach.

That is really not a math problem but more in line [pun intended] with topology.


The sextant/cocked hat/LOP problem really devolves down to more complex 
questions as to what the multiple error factors are and might be, and 
probably is equally trivial aside from the number crunching and research 
needed to determine the error factors and quantify them. Theories are all 
well and fine, but theories as to errors are often put to the lie by simple 
field experiments. Someone needs to go out, fudge up their readings to 
accomodate the entire range of potential bumbling errors, and then see how 
that affects the plot versus the real location.

And I'm betting that the most reliable answer will still be "The center of the 
cocked hat, overlaid with a suitable circle of uncertainty". Most of the 
navigational skill lies not in plotting the cocked hat, but in accurately 
sizing the circle of error.

   

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