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<title>GeographicLib: Transverse Mercator Projection</title>
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<h1>Transverse Mercator Projection </h1> </div>
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<div class="contents">
<center> Back to <a class="el" href="organization.html">Code organization</a>. Forward to <a class="el" href="geodesic.html">Geodesics on the Ellipsoid</a>. Up to <a class="el" href="index.html#contents">Contents</a>. </center><p><a class="el" href="classGeographicLib_1_1TransverseMercator.html" title="Transverse Mercator Projection.">GeographicLib::TransverseMercator</a> and <a class="el" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">GeographicLib::TransverseMercatorExact</a> provide accurate implelmentations of the transverse Mercator projection. The <a href="TransverseMercatorTest.1.html">TransverseMercatorTest</a> utility provides an interface to these classes.</p>
<p>References</p>
<ul>
<li>L. Krüger, <a href="http://dx.doi.org/10.2312/GFZ.b103-krueger28">Konforme Abbildung des Erdellipsoids in der Ebene</a> (Conformal mapping of the ellipsoidal earth to the plane), Royal Prussian Geodetic Institute, New Series 52, 172 pp. (1912).</li>
<li>L. P. Lee, Conformal Projections Based on Elliptic Functions, (B. V. Gutsell, Toronto, 1976), 128pp., ISBN: 0919870163 (Also appeared as: Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13). Part V, pp. 67–101, <a href="http://dx.doi.org/10.3138/X687-1574-4325-WM62">Conformal Projections Based On Jacobian Elliptic Functions</a>.</li>
<li>C. F. F. Karney, <a href="http://dx.doi.org/10.1007/s00190-011-0445-3">Transverse Mercator with an accuracy of a few nanometers,</a> J. Geodesy (2011); preprint <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>; resource page <a href="http://geographiclib.sf.net/tm.html">tm.html</a>.</li>
</ul>
<p>The algorithm for <a class="el" href="classGeographicLib_1_1TransverseMercator.html" title="Transverse Mercator Projection.">GeographicLib::TransverseMercator</a> is based on Krüger (1912); that for <a class="el" href="classGeographicLib_1_1TransverseMercatorExact.html" title="An exact implementation of the Transverse Mercator Projection.">GeographicLib::TransverseMercatorExact</a> is based on Lee (1976).</p>
<h2><a class="anchor" id="testmerc"></a>
Test data for the transverse Mercator projection</h2>
<p>A test set for the transverse Mercator projection is available at</p>
<ul>
<li><a href="http://sf.net/projects/geographiclib/files/testdata/TMcoords.dat.gz/download">TMcoords.dat.gz</a></li>
</ul>
<p>This is about 17 MB (compressed). This test set consists of a set of geographic coordinates together with the corresponding transverse Mercator coordinates. The WGS84 ellipsoid is used, with central meridian 0<sup>o</sup>, central scale factor 0.9996 (the UTM value), false easting = false northing = 0 m.</p>
<p>Each line of the test set gives 6 space delimited numbers</p>
<ul>
<li>latitude (degrees, exact)</li>
<li>longitude (degrees, exact — see below)</li>
<li>easting (meters, accurate to 0.1 pm)</li>
<li>northing (meters, accurate to 0.1 pm)</li>
<li>meridian convergence (degrees, accurate to 10<sup>-18</sup> deg)</li>
<li>scale (accurate to 10<sup>-20</sup>)</li>
</ul>
<p>The latitude and longitude are all multiples of 10<sup>-12</sup> deg and should be regarded as exact, except that longitude = 82.63627282416406551 should be interpreted as exactly 90 (1 - <em>e</em>) degrees. These results are computed using Lee's formulas with <a href="http://en.wikipedia.org/wiki/Maxima_(software)">Maxima</a>'s bfloats and fpprec set to 80 (so the errors in the data are probably 1/2 of the values quoted above). The Maxima code, <a href="tm.mac">tm.mac</a> and <a href="ellint.mac">ellint.mac</a>, used to prepare this data set is included in the distribution. You will need to have Maxima installed to use this code. The comments at the top of <a href="tm.mac">tm.mac</a> illustrate how to run it.</p>
<p>The contents of the file are as follows:</p>
<ul>
<li>250000 entries randomly distributed in lat in [0, 90], lon in [0, 90];</li>
<li>1000 entries randomly distributed on lat in [0, 90], lon = 0;</li>
<li>1000 entries randomly distributed on lat = 0, lon in [0, 90];</li>
<li>1000 entries randomly distributed on lat in [0, 90], lon = 90;</li>
<li>1000 entries close to lat = 90 with lon in [0, 90];</li>
<li>1000 entries close to lat = 0, lon = 0 with lat >= 0, lon >= 0;</li>
<li>1000 entries close to lat = 0, lon = 90 with lat >= 0, lon <= 90;</li>
<li>2000 entries close to lat = 0, lon = 90 (1 - <em>e</em>) with lat >= 0;</li>
<li>25000 entries randomly distributed in lat in [-89, 0], lon in [90 (1 - <em>e</em>), 90];</li>
<li>1000 entries randomly distributed on lat in [-89, 0], lon = 90;</li>
<li>1000 entries randomly distributed on lat in [-89, 0], lon = 90 (1 - <em>e</em>);</li>
<li>1000 entries close to lat = 0, lon = 90 (lat < 0, lon <= 90);</li>
<li>1000 entries close to lat = 0, lon = 90 (1 - <em>e</em>) (lat < 0, lon <= 90 (1 - <em>e</em>));</li>
</ul>
<p>(a total of 287000 entries). The entries for lat < 0<sup>o</sup> and lon in [90<sup>o</sup> (1 - <em>e</em>), 90<sup>o</sup>] use the "extended" domain for the transverse Mercator projection explained in Sec. 5 of <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>. The first 258000 entries have lat >= 0<sup>o</sup> and are suitable for testing implementations following the standard convention.</p>
<h2><a class="anchor" id="tmseries"></a>
Series approximation for transverse Mercator</h2>
<p>Krüger (1912) gives a 4th-order approximation to the transverse Mercator projection. This is accurate to about 200 nm within the UTM domain. Here we present the series extended to 10th order. By default, <a class="el" href="classGeographicLib_1_1TransverseMercator.html" title="Transverse Mercator Projection.">GeographicLib::TransverseMercator</a> uses the 6th-order approximation. The preprocessor variable TM_TX_MAXPOW can be used to select an order from 4 thru 8. The series expanded to order <em>n</em><sup>30</sup> are given in <a href="tmseries30.html">tmseries30.html</a>.</p>
<p>In the formulas below ^ indicates exponentiation (<em>n^3</em> = <em>n*<em>n*<em>n</em>)</em> and</em> / indicates real division (3/5 = 0.6). The equations need to be converted to Horner form, but are here left in expanded form so that they can be easily truncated to lower order in <em>n</em>. Some of the integers here are not representable as 32-bit integers and will need to be included as double literals.</p>
<p><em>A</em> in Krüger, p. 12, eq. (5) </p>
<div class="fragment"><pre class="fragment">
A = a/(n + 1) * (1 + 1/4 * n^2
+ 1/64 * n^4
+ 1/256 * n^6
+ 25/16384 * n^8
+ 49/65536 * n^10);
</pre></div><p><em>gamma</em> in Krüger, p. 21, eq. (41) </p>
<div class="fragment"><pre class="fragment">
alpha[1] = 1/2 * n
- 2/3 * n^2
+ 5/16 * n^3
+ 41/180 * n^4
- 127/288 * n^5
+ 7891/37800 * n^6
+ 72161/387072 * n^7
- 18975107/50803200 * n^8
+ 60193001/290304000 * n^9
+ 134592031/1026432000 * n^10;
alpha[2] = 13/48 * n^2
- 3/5 * n^3
+ 557/1440 * n^4
+ 281/630 * n^5
- 1983433/1935360 * n^6
+ 13769/28800 * n^7
+ 148003883/174182400 * n^8
- 705286231/465696000 * n^9
+ 1703267974087/3218890752000 * n^10;
alpha[3] = 61/240 * n^3
- 103/140 * n^4
+ 15061/26880 * n^5
+ 167603/181440 * n^6
- 67102379/29030400 * n^7
+ 79682431/79833600 * n^8
+ 6304945039/2128896000 * n^9
- 6601904925257/1307674368000 * n^10;
alpha[4] = 49561/161280 * n^4
- 179/168 * n^5
+ 6601661/7257600 * n^6
+ 97445/49896 * n^7
- 40176129013/7664025600 * n^8
+ 138471097/66528000 * n^9
+ 48087451385201/5230697472000 * n^10;
alpha[5] = 34729/80640 * n^5
- 3418889/1995840 * n^6
+ 14644087/9123840 * n^7
+ 2605413599/622702080 * n^8
- 31015475399/2583060480 * n^9
+ 5820486440369/1307674368000 * n^10;
alpha[6] = 212378941/319334400 * n^6
- 30705481/10378368 * n^7
+ 175214326799/58118860800 * n^8
+ 870492877/96096000 * n^9
- 1328004581729009/47823519744000 * n^10;
alpha[7] = 1522256789/1383782400 * n^7
- 16759934899/3113510400 * n^8
+ 1315149374443/221405184000 * n^9
+ 71809987837451/3629463552000 * n^10;
alpha[8] = 1424729850961/743921418240 * n^8
- 256783708069/25204608000 * n^9
+ 2468749292989891/203249958912000 * n^10;
alpha[9] = 21091646195357/6080126976000 * n^9
- 67196182138355857/3379030566912000 * n^10;
alpha[10]= 77911515623232821/12014330904576000 * n^10;
</pre></div><p><em>beta</em> in Krüger, p. 18, eq. (26*) </p>
<div class="fragment"><pre class="fragment">
beta[1] = 1/2 * n
- 2/3 * n^2
+ 37/96 * n^3
- 1/360 * n^4
- 81/512 * n^5
+ 96199/604800 * n^6
- 5406467/38707200 * n^7
+ 7944359/67737600 * n^8
- 7378753979/97542144000 * n^9
+ 25123531261/804722688000 * n^10;
beta[2] = 1/48 * n^2
+ 1/15 * n^3
- 437/1440 * n^4
+ 46/105 * n^5
- 1118711/3870720 * n^6
+ 51841/1209600 * n^7
+ 24749483/348364800 * n^8
- 115295683/1397088000 * n^9
+ 5487737251099/51502252032000 * n^10;
beta[3] = 17/480 * n^3
- 37/840 * n^4
- 209/4480 * n^5
+ 5569/90720 * n^6
+ 9261899/58060800 * n^7
- 6457463/17740800 * n^8
+ 2473691167/9289728000 * n^9
- 852549456029/20922789888000 * n^10;
beta[4] = 4397/161280 * n^4
- 11/504 * n^5
- 830251/7257600 * n^6
+ 466511/2494800 * n^7
+ 324154477/7664025600 * n^8
- 937932223/3891888000 * n^9
- 89112264211/5230697472000 * n^10;
beta[5] = 4583/161280 * n^5
- 108847/3991680 * n^6
- 8005831/63866880 * n^7
+ 22894433/124540416 * n^8
+ 112731569449/557941063680 * n^9
- 5391039814733/10461394944000 * n^10;
beta[6] = 20648693/638668800 * n^6
- 16363163/518918400 * n^7
- 2204645983/12915302400 * n^8
+ 4543317553/18162144000 * n^9
+ 54894890298749/167382319104000 * n^10;
beta[7] = 219941297/5535129600 * n^7
- 497323811/12454041600 * n^8
- 79431132943/332107776000 * n^9
+ 4346429528407/12703122432000 * n^10;
beta[8] = 191773887257/3719607091200 * n^8
- 17822319343/336825216000 * n^9
- 497155444501631/1422749712384000 * n^10;
beta[9] = 11025641854267/158083301376000 * n^9
- 492293158444691/6758061133824000 * n^10;
beta[10]= 7028504530429621/72085985427456000 * n^10;
</pre></div><p>The high-order expansions for <em>alpha</em> and <em>beta</em> were produced by the Maxima program <a href="tmseries.mac">tmseries.mac</a> (included in the distribution). To run, start Maxima and enter </p>
<div class="fragment"><pre class="fragment">
load("tmseries.mac")$
</pre></div><p> Further instructions are included at the top of the file.</p>
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