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    Dip formula derivation
    From: Paul Hirose
    Date: 2001 Jun 22, 2:46 PM

    Yesterday I said the following formula, adapted from one I found in a
    surveying textbook, gives the rise of a horizontal line of sight above
    a level surface over a known distance:
    rise = 6.75e-8 * d * d
    I said it would work for any units, i.e., for distance in feet, rise
    is in feet. That was wrong. The formula is correct for meters only.
    A little thought experiment will prove that formula can't possibly
    work for all units. Suppose it's correct for yards. We compute rise
    over a certain distance. Now do it again, same distance but expressed
    in feet. The answer should be 3x bigger, but since distance is squared
    on the right side, the number you actually get is 9x bigger.
    Once I noticed that blunder, it was straightforward to derive a dip
    formula. There are two key ideas: 1) distance to the sea horizon is
    the distance at which "rise" equals height of eye, which implies 2)
    the horizontal line of sight of an observer h meters high is 2 * h
    meters high when it crosses the horizon.
    If the horizon is d meters away, the dip (in radians) to practical
    accuracy is (2*h)/d. We know h (height of eye) and can compute d from
    the surveyor's formula. After cleaning up, the final formula is: dip =
    .02977 * sqrt(h), where dip is in degrees and h is meters.
    The Nautical Almanac formula is .0293 * sqrt(h), about 1.6% different.
    Dip short is a little easier. For practical purposes, dip short (in
    radians) equals (rise + h) / d. We already know h and d. Rise is easy
    to calculate. In units of arc minutes, feet and nautical miles:
    dip short = .4298 * d + .5658 * h / d
    Compared to the table in Bowditch, this agrees within 0.2'. I am now
    happy. Distance to the horizon, geographic range, dip, and dip short
    all stem from one equation.
    paulhirose@earthlink.net (Paul Hirose)

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