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      Addenda for Transverse Mercator with an accuracy of a few nm
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    <meta name="author" content="Charles F. F. Karney" />
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    <a href="tm.html">Back to resource page for transverse Mercator.</a>
    <h3>Addenda for C. F. F. Karney,
      <a href="tm.html">
	<i>Transverse Mercator with an accuracy of a few nanometers</i></a>.
    </h3>
    <p>Addenda for
      <a href="https://doi.org/10.1007/s00190-011-0445-3">
	J. Geodesy <b>85</b>(8), 475&ndash;485 (Aug. 2011)</a> and
      <a href="https://arxiv.org/abs/1002.1417v3">
	arXiv:1002.1417v3 (Feb. 2011)</a>:
      <ul>
	<li>
	  A better starting guess for Newton's method for solving for
	  the &tau; in terms of &tau;', Eqs.&nbsp;(19)&ndash;(21), is
	  &tau;<sub>0</sub>&nbsp;= &tau;'/(1&nbsp;-&nbsp;<i>e</i><sup>2</sup>)
	  in Eq.&nbsp;(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 |&phi;|&nbsp;&lt;
	  3.35&deg;.
	<li>
	  A good way to visualize the transverse Mercator projection
	  over the entire global is using
	  <a href="tm-grid.kmz"
	     type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>,
	  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 <i>x</i> and <i>y</i> directions.  The
	  scale, <i>k</i> = 0.9998035, has been adjusted so that the
	  distance from the equator to a pole is 10000 km.<br>  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.)
	<li>
	  Color versions of the figures in the papers are available in
	  vector format in the PDF file <a href="tm-figs.pdf"
	  type="application/pdf"> tm-figs.pdf</a> or in PNG format
	  <a href="html/transversemercator.html#tmfigures"> here</a>.
	<li>
	  Google Books does not consistently provide access to the full
	  text.  If you encounter this situation, you can download the
	  pdf file listed here:
	  <ul>
	    <li>
	      Lambert (1772),
	      Google id:
	      <a href="https://books.google.com/books?id=o_s_MR3NUD4C">
		o_s_MR3NUD4C</a>,
	      pdf:
	      <a href="https://geographiclib.sourceforge.io/geodesic-papers/lambert72.pdf">
		lambert72.pdf</a>.
	  </ul>
      </ul>
    </p>
    <a href="tm.html">Back to resource page for transverse Mercator.</a>
    <hr>
    <address>Charles Karney
      <a href="mailto:charles@karney.com">&lt;charles@karney.com&gt;</a>
      (2017-09-30)</address>
    <br>
    <a href="https://geographiclib.sourceforge.io">
      GeographicLib home
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