[OSRS-PROJ] Transverse Mercator revisited

Gerald I. Evenden gerald.evenden at verizon.net
Tue Oct 14 21:10:09 EDT 2003

[gie at localhost dozier]$ cat biblio.txt
Tomorrow I will truck down to my modest local library and
visit the reference desk hoping I can get a copy of references
[2] through [4] below.  This process normally take 30+ days
IF successful.  I'm from Missouri and I want some solid
proof and not allegations.

According to Dozier, the problem is quite simple as I have
summarized at the end of this email.

The following is a biblography that appears to be related to
the problem of the "global" transverse mercator projection:

[1] Dozier, Jeff, 1980, "Improved Algorithm for Calculation
of UTM and Geodetic Coordinates": NOAA Technical Report NESS 81,
19 p.

[2] Lee, L.P., 1976, "Conformal projections based on
elliptical functions":---this shows up under two publishers:
B.V. Gutsell (Canada, I believe) and Canadian Cartographer
Monograph 16, Toronto: University of Toronto.

[3] Lee, L.P., 1962, "The transverse mercator projection
of the entire spheroid": Empire Survey Review, 16, 208-217.

[4] Snyder, J.P., 1979, "Calculating Map Projections for the
Ellipsoid": The American Cartographer, Vol. 6, No. 1.

Strebe makes reference to a "Wallis" however I cannot locate
any bib reference to this name except for a reference to
a Helen Wallis and a Davis Wallis award for cartographic
excellence. (?)

Incidently, in email communications with Dozier, he claimed
Snyder coded the process of his paper into a pocket
calculator (probably a TI that Snyder loved to work on).

In Dozier's paper, reference [2] was claimed to be the source
of the base equations for the transverse mercator.  The
general expression transverse mercator projection of a

psi + i lambda = arctanh(sn w) - e arctanh(e sn w)  (1)

psi = isometic latitude
lambda = longitude (from CM)

e = eccentricity

w = u + i v

sn = one of the Jacobian functions.

The Gauss-Kruger class is defined as:

x + iy = (1-e^2)int_0^w dn^-2t dt

    = E(w|e^2) - e^2 (sn w cn w)/dn w     (2)

where E is the elliptic integral of the second kind and
cn and dn are additional Jacobian functions.  i is
square root of -1.  Note x is northing and y easting.

Dozier's statement of the problem is: if you can find w
from equation 1 you can compute x&y from equation 2.
For equation 1 he uses Newton-Raphson and the rest is
simple, eh?
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