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    Re: Snellius Construction questions
    From: George Huxtable
    Date: 2009 Jan 6, 11:53 -0000

    Todd Frye wrote-
    
    I have found an intriguing example of triangulation from a 16th century
    mathematician, Willebrord
    Snellius(http://www.sailingissues.com/navcourse4.html). I have searched
    other web sources on the web and so far, have had no success in finding a
    straight forward explanation of how this method of triangulation works. Have
    any of you actually used the "Snellius Construction" technique, and could
    you help me understand it as a practical navigation application?
    
    ===================
    
    If the name Willebrord Snellius has a familiar ring, that's because it's the
    Latinised version of the name of the discoverer of "Snell's Law", governing
    the refraction of light across a boundary between two media.
    
    The construction that Todd refers to was being taught as a standard method
    in evening-classes in coastal navigation in the UK, all of 50 years ago,
    though never attributed to Snell. You will find it treated in Bowditch, as a
    "three-point fix"; in my 1977 edition it's in chapter XI of vol 1. He also
    refers to a special tool for the job, as a "three-arm protractor", which in
    the UK is more usually called a "station pointer". The "poor man's"
    alternative is simply to draw, on a piece of tracing paper, three converging
    lines, meeting at a point, the angles between those lines corresponding to
    the angles between the landmarks. Then slide that tracing paper over the
    chart, until the lines pass through all three marks, and you must be at
    their crossing-point.
    
    But this thee-point fix is really a tool for coastal surveying, offering
    high precision, because  the relative horizontal angles between the
    landmarks can be measured with a sextant, much more precisely than with a
    compass; however, that level of precision is seldom necessary in coastal
    navigation.
    
    The opportunity of finding three identifiable, well-charted, landmarks
    seldom offers itself until you are rather close inshore, and if you're in
    motion, calls for the angles to be measured simultaneously, or nearly so.
    And when close inshore, by the time you've measured the angles and done the
    necessary plotting, you've moved on, so the precision is no longer relevant.
    That's why it's more useful to hydrographic surveyors, working to high
    accuracy from a stationary vessel.
    
    In practical coastal navigation, my preferred approach is the traditional
    one by compass, as follows- as soon as one landmark or light has been picked
    up and identified, draw a position line on your chart by a compass bearing.
    To do that, of course, you have to be confident about compass corrections,
    but then most navigators are, unless they are in unfamiliar waters or an
    unfamiliar craft. As soon as you identify a second object, draw in both
    bearings, and you have a fix. When you pick up a third landmark, draw in all
    three, to give a triangle or cocked hat. Only if that is unfeasibly large
    should you then worry.
    
    There's  a problem in using the 3-point method, with relative bearings. If
    all three landmarks happen to lie on or nearly-on the same circle (which you
    don't usually realise until you've made the observations and tried to plot
    them) then the two position circles (for which you need the point of
    intersection) coalesce.  That means that you could be anywhere on those
    circles; there's no way of distinguishing where, and all the precision has
    been lost in the resulting ambiguity.
    
    George.
    
    contact George Huxtable, at  george{at}hux.me.uk
    or at +44 1865 820222 (from UK, 01865 820222)
    or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    
    
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