In 1989 Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to program but retain millimeter accuracy. Bowring rewrote the fourth order Redfearn series (after discarding small terms) in a more compact notation by replacing the spherical terms, i.e. those independent of ellipticity, by the exact expressions used in the spherical transverse Mercator projection. There was no gain in accuracy since the elliptic terms were still truncated at the 1mm level. Such modifications were of possible use when computing resources were minimal.
Contents
Notation
E = distance east of the central meridian, measured on the Transverse Mercator projection
N = distance north of the equator, measured on the Transverse Mercator projection
where r is the reciprocal of the flattening for the chosen spheroid (for WGS84, r = 298.257223563 exactly).
(Note that
Transverse Mercator to Lat-Lon
To convert Transverse Mercator coordinates to lat-lon, first calculate
(
Meridian distance
Bowring gave formulas for meridian distance (the distance from the equator to the given latitude along a north-south line on the spheroid) that seem to be correct within 0.001 millimeter on earth-size spheroids. The symbol n is the same as in the Redfearn formulas
Discard the real part of the complex number Z; subtract the real coefficient of the imaginary part of Z from
(Note that if latitude is 90 degrees, then
For the inverse (given meridian distance, calculate latitude), calculate
Discard the real part of Z' and add the real coefficient of i to
If Zero is not at the Equator
As given above, all the formulas for the ellipsoid assume that the Northing on the Transverse Mercator projection starts from zero at the Equator, as it does in the northern-hemisphere UTM projection. People using the British National Grid, or State Plane Coordinates in the United States, have an additional step in their calculations.
The British National Grid sets Northing at (latitude 49 degrees North, longitude 2 degrees West) to be -100,000 meters exactly. It uses the Airy spheroid, with equatorial radius being 6377563.39603 meters and the reciprocal of the flattening being 299.3249645938 (both values being rounded); the meridian distance from the equator to 49 degrees latitude therefore calculates to 5429228.602 meters on the spheroid. Rounded scale factor at longitude 2 degrees west is 0.999601271775, so on the Transverse Mercator projection 49 degrees North is 5427063.8153 meters from the Equator.
So when converting lat-lon to British National Grid, use the formulas given above and subtract 5527063.815 meters from the calculated N.
Example: convert lat-lon to UTM
NGS says the Washington Monument is 38 deg 53 min 22.08269 sec North, 77 deg 02 min 06.86575 sec West on NAD83; what's its UTM?
As with all NAD83 calculations we use the GRS80 spheroid with a = 6378137 meters exactly and r = 298.25722 2101 rounded. If we lazily take that value of r as exact we get
z is -1.43831 52572 times
Next get m, the meridian distance from the equator to the monument:
so p = 0.99972936, q = 0.00122999 and the imaginary part of Z is 0.000820069 times i.
Subtract 0.000820069 from 0.677108669 to get
Plug all of that in and we get N = 4306479.5101 meters, E = -176516.8552 meters; add the latter to 500000 (the Easting value along the central meridian in all UTM zones) to get UTM Easting of 323483.1448 meters, which agrees with the NGS datasheet.