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    Re: Coastal Plotting Sheets
    From: Peter Fogg
    Date: 2007 Apr 1, 16:25 +1000

    On 3/28/07, Michael Bradley wrote:
    Given a known angle between two known objects viewed
    from the boat,
    it is possible to construct a circular LOP for the
    boat through the two objects.
    This is then repeated for another pair of objects to
    get another circular LOP. Where these
    circles cut is the boat's position. Because you save
    the geometric effort of drawing the circles when using
    a station pointer or similar, you get no indication of
    the angle of cut between the two circles, hence the
    potential danger. 

    To do the geometry, you draw a 'base line' between the
    two known objects, take 90 deg minus the observed
    angle = x, draw a line from each object at angle x to
    the base line. Where these two lines meet is the
    centre of the circle, which you can then draw with
    your compasses. If your drawing is accurate, the
    circle passes through the fixed objects. The boat lies
    somewhere on the circle which I've called the circular
    LOP. Repeat for another pair of objects and their
    particular angular difference to construct the second
    circular LOP. The circular LOPs usually cross, but
    there's no certainty that they will...

    In the extreme case, if all three fixed objects lie on
    a position circle which also crosses the boat's
    position, the 'plotted' position circles will lay over
    each other, and the boat will get the same ambiguous
    horizontal angles wherever it lies on the joint
    position circle -  a major danger.

    The general advice to avoid these dangers is to ensure
    that the middle object used for the fixes is the one
    nearest to the boat. In that case there is no way the
    boat can be on a circular LOP which includes all three
    objects. Even so, that set up does not ensure a decent
    angle of cut.

    Michael, I drew up your example (thanks for the sketch - with which program did you draw it?) with shore points A, B & C lying along a straight line (eg; all on the edge of a straight coast) and that worked out fine, although I noticed that the LOP is actually only an arc of a circle; the boat cannot be 'behind' the shore objects nor can it be closer to the 'coast' than two points along the circle that still allows the horizontal angle between the two points to be achieved.

    Then drew it again with B well 'inland' of A and C. I ended up with a smaller circle wholly within the larger; there was no common point at which the two angles could have been observed. If this was the case then the possibility of an erroneous position being derived would not arise.

    Then drew one of the circular LOPs onto clear plastic and moved it around the other circle, effectively bringing point B closer to the shoreline, noting that indeed the two circles could effectively overlap or lie close to each other : the poor 'cut' you referred to.

    Presumably this technique would be useful if a compass was not available but horizontal angles between shore objects could be measured, with a sextant for example, although the distance inland of the middle object was unknown. Then the danger you warn of would be relevant.

    However, when using corrected compass bearings direct and unambiguous LOPs are derived that can be plotted directly onto the chart from the shore points seawards, leading to an intersecting fix from two points or an encompassing shape indicating the fix from three points or more.

    The alternative of plotting from a point shore-wards, whether with a station pointer or with lines drawn on clear material, leads to an unambiguous fix when the angles have been established via use of a compass. Even if the position of a shore object relative to the coast (a presumed straight line linking all three) was unknown, since these angles have been derived from compass observations these corrected compass bearings provide an effective reality check. The fix is unambiguous.

    Thanks for providing the impetus for better understanding this method of drawing up a position from horizontal angles only of shore objects, and the shortcomings of the method.

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