# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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