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    The One In Sixty Rule (was: Re: going overboard on decimals)
    From: Thomas Schmidt
    Date: 1999 Jul 28, 9:07 AM

    Millard Kirk wrote:
    ...
    > After the GPS
    > showed a bearing of 1 degree difference from our course I was wondering how
    > much off course that represented, without plotting of course.   He made a
    > statement that we were only 1 nautical mile off course.  I was able to
    > understand from his discussion that may be true  if one is at least 60 miles
    > from his destination one (1) degree would calculate about that.  I have
    > since run across a rule "The One In Sixty Rule" although not fully
    > explained, but I think it is what he was referring to.
    ...
    
    
    Being a hopeless landlubber, I am not familiar with these rules, but it
    would make sense.
    
    If your two courses (ideal and real) differ by one degree, then the
    distance d between them at a distance D from where they intersect (be
    that your destination or your departure point) is
    
     d = D * tan( 1 deg )
       = D * 0.0175
       = D / 57.3
    
    which roughly translates to an error which is about one sixtieth
    of the distance i.e.,
      one km at 60 km distance or
      one nm at 60 nm distance or
      two nm at 120 nm distance etc.
    
    
    Furthermore, since we generally have as an approximation for small
    angles alpha (in degrees)
    
     tan( alpha ) =  alpha * 3.1415/180 = alpha * 0.017453 = alpha / 57.296
    
    the above rule generalizes to
    
     d = D * tan( alpha )
       = D * alpha / 57.296
    
    So for a course deviation of 2 degrees, we have an error of two
    sixtieths of the distance; for three degrees, three sixtieths
    of the distance etc.
    
    This breaks down if the angles become too large since we used an
    approximation which is valid for small angles only. The exact
    numbers would be:
    
      1 degree  off:   d = D *  1 / 57.290
      2 degrees off:   d = D *  2 / 57.273
      3 degrees off:   d = D *  3 / 57.243
      4 degrees off:   d = D *  4 / 57.203
      5 degrees off:   d = D *  5 / 57.150
     10 degrees off:   d = D * 10 / 56.713
     20 degrees off:   d = D * 20 / 54.950
     30 degrees off:   d = D * 30 / 51.962
     40 degrees off:   d = D * 40 / 47.670
     etc.
    
    
    So as a rule of thumb, if you are one degree off course and at a
    distance of 60 nm from your destination, your lateral displacement
    is 1 nm. For other distances, adjust the result proportionately.
    For other course deviations, multiply with the number of degrees
    you are off.
    You get your result in the same length units you used for the distance
    (nautical miles or kilometers or fathoms... :)
    
    The rule starts to break down when your course deviation becomes too
    large (because then '60' is no good approximation any more) and when
    the distance becomes too large (because then the above formulas from
    plane trigonometry cease to be applicable).
    
    Sorry if the above lacks the appropriate lingo, but my Bowditch is
    at home...
    
    --
      -------------------------------------------------------------------
      Thomas Schmidt                    e-mail  : schmidt{at}hoki.ibp.fhg.de
    

       
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