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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 Browse Files

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