A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: David Pike
Date: 2019 May 2, 14:30 -0700
Bill you wrote:
The fun thing from the geometry point of view again (Oh no Professor Lionheart is going on about triangles again!)
If the only error is a constant error for each of three lines of position the true position is the incentre of the cocked hat, or one of the excentres, depending on which side of the lines of position the GP of the body is on.
(Image from Wikipedia see https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle#/media/File:Incircle_and_Excircles.svg)
I think you can argue it out using words without going too deeply into Wikipedia. If the three angles between the three azimuths are all less than 180°, you know where you're starting from, and there is absolutely no error, then the fix is where the azimuths cross. As the error increases at the same rate for each PL, they will all depart from the centre spot by the same amount, so the three equal errors will each be a radius of the inscribed circle.
I must admit, I’d never heard of the excentres until tonight, so it’s really one for your students, but if you start with the three stars all on the same side of the ship or aircraft, then it doesn’t take much scratching around on the back of a envelope to see that as the PLs move away at a constant rate from the point where the azimuths cross, the errors once more are three equal radii of an excircle centred at the point where the azimuths cross. DaveP