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    Re: AP terminology, WAS: 2-Body Fix -- take three
    From: Frank Reed
    Date: 2009 Nov 13, 14:29 -0800

    I wrote previously,
    "Well, you can do that today. Historically it was prohibitively inefficient in 
    terms of computation cost. But today if I observe Venus 4d 12.5' above the 
    horizon, I can easily calculate every spot on the Earth where this would be 
    the observed altitude."
    And John you wrote in a later message,
    "No one has addressed my question of why the St Hilaire method calculates an 
    altitude at a location our ship is NOT at, when we've just measured the 
    altitude where our ship IS at."
    Followed later by,
    "So I'll put my question yet another way:  Why is the St Hilaire method
    superior to Sumner's and consequently the only one used today??"
    My reply above did address your question, I think, and the answer to your 
    latest version is the same. It's computationally "cheaper" to do it by St 
    Hilaire. This mattered a great deal 25 to 100 years ago and the increase in 
    computational efficiency eventually (very slowly!) made these methods almost 
    the only ones used in the latter half of the 20th century, but that 
    compuational efficiency matters much less today. In fact there are quite a 
    few people who use the Sumner method to plot points today when the 
    calculations are done by computer. Of course, the traditional Sumner approach 
    of picking a couple of points and drawing a straight line through them is 
    still only an approximation. With the astounding amount of computation power 
    available even on tiny devices like cell phones today, it's quite possible to 
    go way beyond the original Sumner approach and calculate everything at every 
    single point on Earth (at some grid spacing, e.g., every 0.1' of lat, lon) 
    and figure out the apparent altitude at every point with all the small 
    details like oblateness handled correctly in a direct calculation. You could 
    even get really crazy with this and include variations in temperature and 
    pressure from weather service data. Then our plots would really be telling us 
    the locations where the observed altitudes would be most closely matched by 
    reality on the globe. Wouldn't that be fun?!
    Finally, I think all this focus on how to calculate LOPs is fundamentally 
    historical. If I take several sights, the plotted result of my observations 
    should be some sort of error ellipse, or several ellipses with different 
    levels of confidence, taking into account all of the sights statistically. 
    Naturally we can include the LOPs in such a plot (and they provide an 
    important check for gross errors), but the error ellipse should be the 
    primary result. And here we can take a lesson from the software in many GPS 
    receivers. When they display the position as a dot, they usually put a shaded 
    circle around it indicating the degree of accuracy you should expect from the 
    satellite fix. When fewer satellites are available, the error circle is 
    larger. More satellites, smaller circle. Of course there are plenty of 
    software packages available for celestial navigation sights that do exactly 
    this, too. Whether there's any convenient way to accommodate such statistical 
    indicators in paper plots with purely manual calculations is a more open 
    issue. The beginning of any such procedure is to locate the center of the 
    error ellipse from three or more LOPs. There is a long-established algorithm 
    for this. Herbert Prinz demonstrated a proof of a clever "straight-edge and 
    dividers" method for finding that point during the Navigation Weekend in 
    Mystic in 2008 (www.fer3.com/Mystic2008) without doing any trig or using any 
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