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    Re: The cocked hat
    From: George Huxtable
    Date: 2004 Apr 4, 17:09 +0100

    Robert Eno wrote-
    >I was persuaded, some years ago, that the 3 star fix is overrated. Better to
    >take two observations of two stars.  This is not an an original idea on my
    >part: my inspiration comes from a little-known book -- Arctic Air
    >Navigation -- by a former RCAF Air Navigator, Keith Greenaway:. In his book
    >Greenaway writes:
    >"Navigators who use asto continually prefer 2-star fixes with each star
    >sighted twice, rather than 3 star fixes with each star sighted only once.
    >Although an extra sight is required, time is actually saved, because only
    >two stars have to be located, the course setting of the sextant changed only
    >once, and the tables entered for two stars only. It is also easier to detect
    >inaccurate computation or observation as shown in Fig. 27".
    >Figure 27 shows two parallel LOP's spaced very closely and two parallel LOPs
    >at right angles to the former, spaced far apart. The figure beside it shows
    >a classic "cocked hat". The caption to the figure reads: "Comparison of a
    >2-star and a 3-star fix. In the case of the 3-star fix, it is not obvious
    >which sight may be in error, while in the case of the 2-star fix it is
    >immediately apparent which star should be re-sighted or sight computations
    Some comments come to mind here.
    What if Robert had identified a wrong star (for example)? If so, no doubt
    he would make the identical error on his second try, so everything will
    SEEM perfectly consistent, a dangerous state of affairs. His technique
    provides little cross-checking. There are many other likely sources of
    error, which are likely to be exactly repeated if the first measurement is
    simply measured again.
    We have to remember that air navigators live under different constraints to
    those of us who sail. They are travelling so fast that unless they can
    produce a quick answer, it may already be too late. On the other hand,
    those of us that sail small vessels usually have plenty of time to put the
    kettle on for a cup of tea, before we get round to tackling sight
    Why do we take a number of star observations, in that short twilight at
    dawn and dusk? Well, partly because they are there, and we can, and they
    can provide a good instantaneous fix. In the daytime, we have to use the
    Sun, and as there is unfortunately only one Sun, we have to make do with
    that, using transferred position lines as it moves round the sky.
    But there's a more subtle reason. Generally speaking, the biggest error in
    an altitude observation is not in our sight of the body itself, but in the
    horizon, that we measure it up from. Except in a small vessel in heavy
    weather, the biggest unknown in the position of the horizon is the dip.
    Part of this dip is just due to geometry, the curvature of the Earth and
    the height of the observer's eye, and that presents no problem. But another
    part comes from the bending of light rays as they skim just above the sea
    from the horizon to the eye, which can be greatly influenced by temperature
    gradients close to the sea surface. It can vary from its normal value by 1,
    2, or occasionally several minutes of arc, in either direction. When this
    is bad, we call it "anomalous dip". If there's a ship to be seen on the
    horizon, its image may be distorted by a "mirage" effect in an extreme
    case, but often there is no way to know that anomalous dip is occurring.
    However, anomalous dip is usually the same at all azimuths, so if it's
    possible to observe three stars with azimuths 120 degrees apart (or nearly
    so), the intercepts will be affected in such a way that the centre of the
    resulting cocked-hat will not be displaced by anomalous dip. It's not
    possible to treat Sun altitudes in that way.
    This is exactly contrary to Joel's argument, in which he said-
    >As everyone knows, the
    >horizon at sea's definition varies with the direction you are looking due to
    >differences in light, sea conditions and cloud cover. Therefore, you have
    >variables with which Greenway is not confronted since his horizon is a
    The reasoning I have presented above relies on the SAMENESS of the
    behaviour of the atmosphere in different directions, not on its DIFFERENCE.
    Of course, from the air with a bubble-sextant, the dip of the horizon is
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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