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Dear Geoff
Interestingly one of my retired colleagues who had seen the talk
pointed out the same observation for the cocked hat for three
bearings.

There was a discussion of the probability contours (ellipses) for the
problem previously on NavList eg
https://navlist.net/m2.aspx/simple-threebody-fix-puzzle-TomSult-dec-2010-g14787
(I am sure I saw a link to a diagram somewhere too)

Even for more lines the system is still distance vector d= Ax-b where
each row of the linear system Ax=b is a line.  So clearly ||d||^2 is
quadratic (ie a conic section) in x and as ||d||^2>=0 it must be an
ellipse.

I might see if I can cook GeoGebra demo that draws the ellipse - it
would have the charm that you could see how it moves as you change the
triangle with the mouse pointer. If you want to see the Symmedian
point this way just  fire up Geogebra (on your PC, tablet phone etc)
select triangle tool. Draw a triangle (you have to close it) then at
the bottom in the Input window type TriangleCentre[A,B,C,6]  (note the
non-American spelling). Now choose the move/arrow tool and play!

Bill


On 1 March 2017 at 15:38, Geoffrey Butt  wrote:
> Bill (in his video) assumes that errors attached to the three LOPs have a
> symmetrical Gaussian distribution.  This implies that there is a 50% chance
> that the true position lies on one side or the other of the plotted line.
> Crudely, this implies that the probability of the true position lying within
> the 'cocked hat' is 0.5 x 0.5 x 0.5 - ie, a 7 to 1 chance that it lies
> outside.  I recall a discussion some years ago which refined the calculation
> of this probability, but still resulting in a greater chance of the true
> position lying outside the triangle.  Rather than concentrating on the
> niceties of finding the point of highest probability within the triangle,
> how would one approach the more useful specification of a boundary (circle?)
> which defines the inclusion of the true position with some specified
> probability?
>
> Geoff Butt
>
> 



--
Professor of Applied Mathematics
http://www.maths.manchester.ac.uk/bl

   

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