Back to resource page for transverse Mercator.
Addenda for
J. Geodesy 85(8), 475–485 (Aug. 2011) and
arXiv:1002.1417v3 (Feb. 2011):

A better starting guess for Newton's method for solving for
the τ in terms of τ', Eqs. (19)–(21), is
τ_{0} = τ'/(1  e^{2})
in Eq. (19). This reduces the number of iterations
required to from two to one for φ < 3.35°.

A good way to visualize the transverse Mercator projection
over the entire global is using
tmgrid.kmz,
which is a Google Earth KML file showing the transverse
Mercator grid (in red) for the WGS84 ellipsoid with grid
spacing 1000 km in the x and y directions. The
scale, k = 0.9998035, has been adjusted so that the
distance from the equator to a pole is 10000 km.
If you
open the "tmgrid" folder in Google Earth and check on the
"sphericaltransversemercator" subfolder, you will also see
the corresponding spherical transverse Mercator grid (in
yellow) conformally mapped to the WGS84 ellipsoid. (This
doesn't have a constant scale on the central meridian.)

Color versions of the figures in the papers are available in
vector format in the PDF file tmfigs.pdf or in PNG format
here.

Google Books does not consistently provide access to the full
text. If you encounter this situation, you can download the
pdf file listed here:
Back to resource page for transverse Mercator.
Charles Karney
<charles@karney.com>
(20130831)