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How pointy is your hat?

If the angles of a triangle were chosen from a uniform distribution
obtuse triangles are more common. In a fix using three LOPs from
observation of celestial bodies the azimuths are not chosen randomly,
they are chosen from the bodies available for observation and the
navigator's judgements as which to use. So in practice what is the
ration observed in practice of obtuse to acute in practice?

One reason I am asking is that for an acute triangle the weights in a
weighted least squares problem can be adjusted so that the elliptical
probability contour is circular. So if one can increase the accuracy
of some LOPs, by taking n sights with a reduction in variance that
goes as 1/(n-2), one can obtain more equal uncertainty in all
directions. But for obtuse triangles the circumcentre lies outside the
triangle and this makes the required weights negative. In practice I
have seen a lot of obtuse cocked hats.

Bill Lionheart

   

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