Addenda for C. F. F. Karney, Transverse Mercator with an accuracy of a few nanometers.

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 τ₀ = τ'/(1 - e²) 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 tm-grid.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 "tm-grid" folder in Google Earth and check on the "spherical-transverse-mercator" 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 tm-figs.pdf or in PNG format here.