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    Re: dip, dip short, distance off with buildings, etc.
    From: Frank Reed CT
    Date: 2006 Jan 13, 02:52 EST

    I wrote:
    > With a sextant we
    > are  measuring the altitude of point  B from point A so that means that  we
    > know the angle in the triangle at point A,  let's call that  gamma.
    
    Bill, you asked:
    "Which triangle?  The small oblique or  large oblique?"
    
    The only triangle I defined . You have point C  at the center of the
    Earth, point A at height h above the Earth's surface, and  point B some distance
    away at height H above the Earth's surface.Those three  points make a big
    triangle ABC. Our GOAL is to find the angle in the triangle at  point C (since, if
    we multiply that by the Earth's radius, we get the distance).  Now what do we
    measure with our sextant or theodolite? Well, we're at point A,  and we measure
    altitudes above the horizontal which is, by definition, the plane
    perpendicular to side AC of the triangle. Now how is that related to the angle  at point
    A in the big triangle (the angle "CAB")? Clearly that's just 90 degrees  plus
    the measured altitude. So we KNOW the angle CAB. This is equivalent to the
    measured parameter (or at minimum, it's related to the measured parameter by a
    simple relationship). So we know the corner angle at point A (the observer),
    which I named "gamma", and we are seeking the corner angle at point C (the
    center of the Earth), which I named "phi". But, uh-oh, we're stuck with another
     angle --the one at point B. Wait... no we're not. Since it's a simple plane
    triangle, all three angles must add up to 180 degrees. That means that the
    angle  in the big triangle at point B is NECESSARILY equal to 180-(gamma+phi).
    Ok so  far?? If you haven't drawn a picture of this yet, you can't possibly be
    ok here   so please make sure you've got a picture of this. And if you
    don't  want to draw your own picture, see image 488 in the archive (see  below).
    
    Now we are in a position where we can apply the law of sines (the  ordinary
    plane trig law of sines...) to the big triangle ABC. Set it up  as
    sin("angle at A")/(R+H)=sin("angle at B")/(R+h).
    And work  from there. You'll need to remember that
    sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b).
    
    That's enough triangles for now, I  think!
    
    I don't know if anyone will get a kick out of it or not, but if  you like
    this kind of math/physics, I've photographed some of my notes. This is  RAW
    material. You can download it at  www.HistoricalAtlas.com/lunars/refnotes.zip. It's
    about 5.7 megabytes. By the  way, try to bookmark or record these addresses
    with the "HistoricalAtlas" part  intact. Yes, it currently points to one of my
    other addresses "clockwk.com" but  that's changing soon.
    
    -FER
    42.0N 87.7W, or 41.4N  72.1W.
    www.HistoricalAtlas.com/lunars
    
    
    

       
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