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    Re: Frank's formulas, was: Lunars: altitude accuracy
    From: Alexandre Eremenko
    Date: 2004 Nov 2, 20:58 -0500

    Dear Frank,
    On Tue, 2 Nov 2004, Frank Reed wrote:
    > Great. I'm glad you looked into it.
    > We might possibly both be wrong, but the
    > odds are declining. ;-)
    Well, a math argument can be wrong only if there is a blunder
    in it:-) And a blunder in a math argument is usually easier
    to find than in any other argument:-)
    I am ready to type a simple complete derivation of these formulas
    and post it here. If there are people interested to check the
    Another question is the relation of the math argument to
    the real world:-)
    > Note that it is not NECESSARY to ignore refraction.
    But taking refraction into account will lead to more
    complicated formulas. One great advantage of those formulas is
    their simplicity. (Which was very unexpected to me; that's why
    I decided to verify them).
    > Incidentally, I also checked these results numerically,
    > adding refraction in
    > as well, and the conclusions I posted previously hold.
    Which is not surprising, because parallax is responsible for
    the greatest correction in the lunars.
    It is MAINLY because of the parallax that we need these altitudes
    at all. Refraction is only SECOND.
    > We discussed the case of a lunar where the altitudes are
    > each 45 degrees on opposite sides of
    > the zenith and the measured distance is
    > exactly 90 degrees. The difference in azimuth is 180 degrees,
    > and the cleared
    > distance is 89d 21.3'.
    > But suppose
    > that's what you've recorded. What happens?
    > It is interesting that if you clear
    > the distance, you will *still* get the number you're looking for:
    > 89d 21.3'.
    This has a simple mathematical explanation.
    > But in this case, if you were to attempt to extract
    > an actual value for the
    > difference in azimuth, you would find a meaningless number
    > (the intermediate step
    > in the calculation gives a value for cosZ of -1.19).
    This is not totally "meaningless":-)
    Trig functions are well defined for COMPLEX arguments.
    If we admit complex arguments, cos can take ANY value,
    not restricted anymore to the interval [-1,1].
    An important principle says that whenever some identity
    is established for real values of the argument,
    it PERSISTS for all (complex) values.
    In our case, the identity is your formula that says that
    the cleared distance is independent of Moon's altitude
    when the distance is 90 degrees. It "persists", (that
    is remains correct) even in you plug the values of the angles
    which have no geometric meaning.
    This phenomenon was first encountered in physics by Fresnel:
    his formulas (for "complete reflection") gave the right answers,
    even when some intermediate terms had no usual meaning because
    some sin'es were greater than 1.

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