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    Re: UTM to lat/lon formulas
    From: Herbert Prinz
    Date: 2003 Dec 13, 00:56 -0500

    Special thanks to Paul Hirose for pointing out an error in the Expl. Suppl. After
    perusing chapter 4, I find that the problem is actually deeper than a mere
    printing error. In fact, there is a general mix-up of the various curvatures of
    the suface of the spheroid. This may not only affect conversion of geodetic
    lat/lon to UTM coordinates, but also datum conversions,  as well as the
    computation of ECIF positions as required for finding topocentric places of
    celestial objects.
    Already on page 206, in formula 4.22-9, we find the correct formula
        N(phi) = a / Sqrt(1- Sqr(e) * Sqr(sin(phi)))
    The magnitude N(phi) is the radius of curvature in the plane perpendicular to the
    meridian. It is required for the computation of geocentric cartesian coordinates
    from the geodetic spherical ones, such as in formula 4.22-7. The only problem is
    that the explanation in the line preceding formula 4.22-9 says that N(phi) is the
    curvature in the meridian, which is wrong. A reader who ignores that explanation
    and does not care what N(phi) means can safely use 4.22-7 in connection with
    4.22-9 and get the right results.
    Later, on page 210, a formula for the curvature in the meridian is given again.
    This time it is stated as
        N(phi) = a * (1-Sqr(e)) / (1- Sqr(e) * Sqr(sin(phi))) ^ (3/2)
    The second member is indeed the correct term for the curvature of the meridian at
    the given latitude, but it should of course not be named N(phi), lest some
    confusion is prone to arise. Let's call it R(phi), in the following. So, we adopt
        R(phi) = a * (1-Sqr(e)) / (1- Sqr(e) * Sqr(sin(phi))) ^ (3/2)
    Now, it turns out that R(phi) is never needed for the conversion between lat/lon
    and UTM. Whenever the supplement refers to N(phi) in chapter 4.233, equation
    4.22-9 is the correct one to use and page 210 is to be ignored, along with the
    explanation accompanying 4.22-9. But R(phi1), being introduced in formula
    4.233-14 appears exclusively in 4.233-8 in the term N(phi1)/R(phi1), which can
    obviously be reduced to
        NoverR(phi1) = (1-Sqr(e)*Sqr(sin(phi1))) / (1-Sqr(e))
    The same substitution can also be made in the equations given on the web page the
    reference to which I gave earlier.
    Herbert Prinz

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