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    Re: watch as compass
    From: James R. Van Zandt
    Date: 2007 Jul 30, 22:20 -0400

    
     Frank Reed  wrote:
    
    >   Jim, you wrote:
    >   "You don't even need the local time - just a stick and a piece of
    >   level ground. Poke a stick in the ground and mark where the end of its
    >   shadow falls.  Wait.  The shadow will move directly east. "
    >
    >   Yes. For a good portion of the day in most latitudes, the shadow moves
    >   nearly due east, just as you say. But bear in mind that the path of
    >   the shadow is not normally a straight line. Most of the time, in most
    >   latitudes, if you trace the shadow path all through the day, it's a
    >   hyperbola. The shadow starts out at dawn running towards the azimuth
    >   where the Sun is rising (that's one asymptote of the hyperbola). Then
    >   it curves about and, right at noon, the shadow is heading due east
    >   exactly. During the afternoon, the shadow turns slowly until, in the
    >   late afternoon, it is running rapidly away from the azimuth where the
    >   Sun is setting (the other asymptote of the hyperbola). The shadow's
    >   path along the ground is curved. But I should emphasize here that the
    >   deviation of the shadow path from straight "due east" is not large
    >   except close to sunrise and sunset and close to the solstices in
    >   higher latitudes.
    >
    ...
    >   If the observer is in a latitude where the Sun does not
    >   set, the path of the shadow tip is still a conic section. Instead of
    >   being a hyperbola, it is an ellipse. In short, do not use this trick
    >   in high latitudes.
    
    Actually I was just remembering simple descriptions like the one at
    http://www.survivaltopics.com/survival/using-shadows-to-determine-direction/
    I suspected there was more to the story, but had not hit upon the flat
    earth approximation.
    
    Thank you for filling it out and describing the limitations.
    
    It's interesting that the possible paths are hyperbolas and ellipses
    just like the 2-body problem with gravitational fields, except here
    the shadow is slowest near the focus.
    
    I suspect that if you know the time between the two marks and enough
    distance measurements, you can figure out which hyperbola the marks
    are on and get a more accurate estimate of direction.  Of course, by
    the time you've worked that analytic geometry problem you might just
    as well wait until noon when the shadow is moving exactly east.
    
              - Jim Van Zandt
    
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