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    On LOPs
    From: Peter Fogg
    Date: 2002 Apr 13, 22:18 +1000

    George Huxtable seems to have declared open season on navigational sacred cows.
    First it was the noon sight, now it seems to be the turn of the idea of finding a
    positive fix position within the confines of three intersecting LOPs.
    'Well, the meaningfulness of a "cocked hat", whether from land bearings or
    astro sights, is frequently misunderstood. It's a surprising fact that no
    matter how good the navigator, only one time in four will his cocked hat
    embrace his actual position, which is three times more likely to lie
    outside it. This is a universal truth, relying in no more than this
    proposition: that each position line, being the best estimate that can be
    made, is just as likely to lie to the left of the true position as to the
    right. So it's a big fallacy to imagine that the true position must be
    within the cocked hat. Instead of a probability of 100%, it will be just
    Not being able to argue from a position of strong mathematical skills (to put it
    mildly) I'll have to do what I can here with common sense (a valuable check on
    practical matters). It is true that the correct LOP could lie on either side of the
    calculated LOP. Where 3 LOPs form a triangle, while any or none of them may be
    correct, the presence of each 'pulls' the probability of 'correctness' towards
    itself, and where this pull intersects is our fix position.
    Let's put it another way. It may be true that the real position could lie outside
    the triangle. The question is, what good is knowing that to us? Can we plot this
    possible point?
    You want smoke and mirrors?
    Hold out your hands in front of you, fingers apart. Count down from 10 on one hand.
    What do you get to? 6. Very good. Now count up the fingers in the other. 5.
    Excellent. 6+5 =? 11. So you have 11 fingers.
    After that demonstration you'll appreciate I'm going to have a Bex, a cup of tea,
    and a nice lie down; then I'll have a go at calculating  'an angle P from P =
    2*arccos(sqr(sin s*sin (s-coalt)/(sin colat*sin codec)))'
    Don't laugh, I may get there, but could be gone a while (Bex was a brand of strong
    Before then, and while we're still on the subject of LOPs, if there are only 2 then
    any error in calculated azimuth, or even any error in plotting can lead to a
    considerable difference in their intersection, the fix position. This is
    particularly true when they are nearly parallel, which happens when the azimuths
    are similar. In astro navigation, as in life, you have to do the best you can with
    what you've got (clouds could cover nearly all the sky) but another sight from
    another part of the sky (we're talking about star sights here) will give you a
    third LOP that cuts across the other 2 and 'pulls' the fix position a long way -
    from the intersection of the 2 nearly parallel LOPs all the way to close to the
    third, much shorter LOP. Why is it so? Because the fix position is equidistant from
    all 3 lines, so its not at the tip of the long triangle, nor halfway along its
    length, but right down near its base. Draw a few and check.
    Here are a couple of 'special purpose' LOPs that you can plan for and use as the
    sun crosses the sky.
    An LOP that runs parallel to the course of the boat is known as a course line, as
    this is what it checks.
    One that runs at right angles to the course is a speed line, and can be used to
    indicate how far down its track the boat has gone.
    So much fun from just one sun.

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