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 starting guess is the geocentric latitude which, to first order in the flattening, is equal to the conformal latitude.) 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.
• Google Books does not consistently provide access to the full text. If you encounter this situation, you can download the pdf file listed here:
  • Lambert (1772), Google id: o_s_MR3NUD4C, pdf: lambert72.pdf.