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<h1>Geodesics on the Ellipsoid </h1>  </div>
</div>
<div class="contents">
<center> Back to <a class="el" href="transversemercator.html">Transverse Mercator Projection</a>. Forward to <a class="el" href="geocentric.html">Geocentric coordinates</a>. Up to <a class="el" href="index.html#contents">Contents</a>. </center><p><a class="el" href="classGeographicLib_1_1Geodesic.html" title="Geodesic calculations">GeographicLib::Geodesic</a> and <a class="el" href="classGeographicLib_1_1GeodesicLine.html" title="A geodesic line.">GeographicLib::GeodesicLine</a> provide accurate solutions to the direct and inverse geodesic problems. The <a href="Geod.1.html">Geod</a> utility provides an interface to these classes. <a class="el" href="classGeographicLib_1_1AzimuthalEquidistant.html" title="Azimuthal Equidistant Projection.">GeographicLib::AzimuthalEquidistant</a> implements the azimuthal equidistant projection in terms of geodesics. <a class="el" href="classGeographicLib_1_1CassiniSoldner.html" title="Cassini-Soldner Projection.">GeographicLib::CassiniSoldner</a> implements a transverse cylindrical equidistant projection in terms of geodesics. The <a href="EquidistantTest.1.html">EquidistantTest</a> utility provides an interface to these projections.</p>
<p>References</p>
<ul>
<li>F. W. Bessel, <a href="http://dx.doi.org/10.1002/asna.201011352">The calculation of longitude and latitude from geodesic measurements (1825)</a>, Astron. Nachr. 331(8), 852-861 (2010); translated by C. F. F. Karney and R. E. Deakin. Preprint: <a href="http://arxiv.org/abs/0908.1824">arXiv:0908.1824</a>.</li>
<li>F. R. Helmert, <a href="http://geographiclib.sf.net/geodesic-papers/helmert80-en.pdf">Mathematical and Physical Theories of Higher Geodesy, Part 1 (1880)</a>, Aeronautical Chart and Information Center (St. Louis, 1964), Chaps. 5&ndash;7.</li>
<li>J. Danielsen, The Area under the Geodesic, Survey Review 30 (232), 61&ndash;66 (1989).</li>
<li>C. F. F. Karney, <a href="http://arxiv.org/abs/1102.1215">Geodesics on an ellipsoid of revolution</a>, Feb. 2011; preprint <a href="http://arxiv.org/abs/1102.1215">arxiv:1102.1215</a>; resource page <a href="http://geographiclib.sf.net/geod.html">geod.html</a>.</li>
<li>A collection of some papers on geodesics is available at <a href="http://geographiclib.sf.net/geodesic-papers/biblio.html">http://geographiclib.sf.net/geodesic-papers/biblio.html</a></li>
</ul>
<h2><a class="anchor" id="testgeod"></a>
Test data for geodesics</h2>
<p>A test set a geodesics is available at</p>
<ul>
<li><a href="http://sf.net/projects/geographiclib/files/testdata/GeodTest.dat.gz/download">GeodTest.dat.gz</a></li>
</ul>
<p>This is about 39 MB (compressed). This consists of a set of geodesics for the WGS84 ellipsoid. A subset of this (consisting of 1/50 of the members &mdash; about 690 kB, compressed) is available at</p>
<ul>
<li><a href="http://sf.net/projects/geographiclib/files/testdata/GeodTest-short.dat.gz/download">GeodTest-short.dat.gz</a></li>
</ul>
<p>Each line of the test set gives 9 space delimited numbers</p>
<ul>
<li>latitude for point 1, <em>lat1</em> (degrees, exact)</li>
<li>longitude for point 1, <em>lon1</em> (degrees, always 0)</li>
<li>azimuth for point 1, <em>azi1</em> (clockwise from north in degrees, exact)</li>
<li>latitude for point 2, <em>lat2</em> (degrees, accurate to 10<sup>-18</sup> deg)</li>
<li>longitude for point 2, <em>lon2</em> (degrees, accurate to 10<sup>-18</sup> deg)</li>
<li>azimuth for point 2, <em>azi2</em> (degrees, accurate to 10<sup>-18</sup> deg)</li>
<li>geodesic distance from point 1 to point 2, <em>s12</em> (meters, exact)</li>
<li>arc distance on the auxiliary sphere, <em>a12</em> (degrees, accurate to 10<sup>-18</sup> deg)</li>
<li>reduced length of the geodesic, <em>m12</em> (meters, accurate to 0.1 pm)</li>
<li>the area under the geodesic, <em>S12</em> (m<sup>2</sup>, accurate to 1 mm<sup>2</sup>)</li>
</ul>
<p>These are computed using as direct geodesic calculations with the given <em>lat1</em>, <em>lon1</em>, <em>azi1</em>, and <em>s12</em>. The distance <em>s12</em> always corresponds to an arc length <em>a12</em> &lt;= 180<sup>o</sup>, so the given geodesics give the shortest paths from point 1 to point 2. For simplicity and without loss of generality, <em>lat1</em> is chosen in [0<sup>o</sup>, 90<sup>o</sup>], <em>lon1</em> is taken to be zero, <em>azi1</em> is chosen in [0<sup>o</sup>, 180<sup>o</sup>]. Furthermore, <em>lat1</em> and <em>azi1</em> are taken to be multiples of 10<sup>-12</sup> deg and <em>s12</em> is a multiple of 0.1 um in [0 m, 20003931.4586254 m]. This results <em>lon2</em> in [0<sup>o</sup>, 180<sup>o</sup>] and <em>azi2</em> in [0<sup>o</sup>, 180<sup>o</sup>].</p>
<p>The direct calculation uses an expansion of the geodesic equations accurate to <em>f<sup>30</sup></em> (approximately 1 part in 10<sup>50</sup>) and is computed with with <a href="http://en.wikipedia.org/wiki/Maxima_(software)">Maxima</a>'s bfloats and fpprec set to 100 (so the errors in the data are probably 1/2 of the values quoted above).</p>
<p>The contents of the file are as follows:</p>
<ul>
<li>100000 entries randomly distributed</li>
<li>50000 entries which are nearly antipodal</li>
<li>50000 entries with short distances</li>
<li>50000 entries with one end near a pole</li>
<li>50000 entries with both ends near opposite poles</li>
<li>50000 entries which are nearly meridional</li>
<li>50000 entries which are nearly equatorial</li>
<li>50000 entries running between vertices (<em>azi1</em> = <em>azi2</em> = 90<sup>o</sup>)</li>
<li>50000 entries ending close to vertices</li>
</ul>
<p>(a total of 500000 entries). The values for <em>s12</em> for the geodesics running between vertices are truncated to a multiple of 0.1 pm and this is used to determine point 2.</p>
<p>This data can be fed to the <a href="Geod.1.html">Geod</a> utility as follows</p>
<ul>
<li>Direct from point 1: <div class="fragment"><pre class="fragment">
  gunzip -c GeodTest.dat.gz | cut -d' ' -f1,2,3,7 | ./Geod
</pre></div> This should yield columns 4, 5, 6, and 9 of the test set.</li>
<li>Direct from point 2: <div class="fragment"><pre class="fragment">
  gunzip -c GeodTest.dat.gz | cut -d' ' -f4,5,6,7 |
  sed "s/ \([^ ]*$\)/ -\1/" | ./Geod
</pre></div> (The sed command negates the distance.) This should yield columns 1, 2, and 3, and the negative of column 9 of the test set.</li>
<li>Inverse between points 1 and 2: <div class="fragment"><pre class="fragment">
  gunzip -c GeodTest.dat.gz | cut -d' ' -f1,2,4,5 | ./Geod -i
</pre></div> This should yield columns 3, 6, 7, and 9 of the test set.</li>
</ul>
<p>Add, e.g., "-p 6", to the call to Geod to change the precision of the output. Adding "-f" causes Geod to print all 9 fields specifying the geodesic (in the same order as the test set).</p>
<h2><a class="anchor" id="geodseries"></a>
Expansions for geodesics</h2>
<p>We give here the series expansions for the various geodesic integrals valid to order <em>f</em><sup>10</sup>. In this release of the code, we use a 6th-order expansions. This is sufficient to maintain accuracy for doubles for the SRMmax ellipsoid (<em>a</em> = 6400 km, <em>f</em> = 1/150). However, the preprocessor macro GEOD_ORD can be used to select any order up to 8. (If using long doubles, with a 64-bit fraction, the default order is 7.) The series expanded to order <em>f</em><sup>30</sup> are given in <a href="geodseries30.html">geodseries30.html</a>.</p>
<p>In the formulas below ^ indicates exponentiation (<em>f^3</em> = <em>f*<em>f*<em>f</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. These expansions were obtained using the the Maxima code, <a href="geod.mac">geod.mac</a>.</p>
<p>In the expansions below, we have</p>
<ul>
<li><em>alpha</em> is the azimuth</li>
<li><em>alpha<sub>0</sub></em> is the azimuth at the equator crossing</li>
<li><em>lambda</em> is the longitude measured from the equator crossing</li>
<li><em>omega</em> is the spherical longitude</li>
<li><em>sigma</em> is the spherical arc length</li>
<li><em>a</em> is the equatorial radius</li>
<li><em>b</em> is the polar semi-axis</li>
<li><em>f</em> is the flattening</li>
<li><em>e<sup>2</sup></em> = <em>f</em> (2 - <em>f</em>)</li>
<li><em>e'<sup>2</sup></em> = <em>e<sup>2</sup>/</em>(1 - <em>e<sup>2</sup></em>)</li>
<li><em>k<sup>2</sup></em> = <em>e'<sup>2</sup></em> cos<sup>2</sup> <em>alpha<sub>0</sub></em> = 4 <em>eps</em> / (1 - <em>eps</em>)<sup>2</sup></li>
<li><em>n</em> = <em>f</em> / (2 - <em>f</em>)</li>
<li><em>c<sup>2</sup></em> = <em>a<sup>2</sup>/2</em> + <em>b<sup>2</sup>/2</em> (tanh<sup>-1</sup> <em>e</em>)/<em>e</em> </li>
<li><em>ep2</em> = <em>e'<sup>2</sup></em> </li>
<li><em>k2</em> = <em>k<sup>2</sup></em> </li>
</ul>
<p>The formula for distance is</p>
<p>&nbsp; &nbsp; <em>s/<em>b</em> =</em> <em>I1</em>(<em>sigma</em>)</p>
<p>where</p>
<p>&nbsp; &nbsp; <em>I1</em>(<em>sigma</em>) = <em>A1</em> (<em>sigma</em> + <em>B1</em>(<em>sigma</em>))<br/>
 &nbsp; &nbsp; <em>B1</em>(<em>sigma</em>) = sum<sub><em>j</em> = 1</sub> <em>C1<sub>j</sub></em> sin(2 <em>j</em> <em>sigma</em>)</p>
<p>and</p>
<div class="fragment"><pre class="fragment">
A1 = (1 + 1/4 * eps^2
        + 1/64 * eps^4
        + 1/256 * eps^6
        + 25/16384 * eps^8
        + 49/65536 * eps^10) / (1 - eps);
</pre></div><div class="fragment"><pre class="fragment">
C1[1] = - 1/2 * eps
        + 3/16 * eps^3
        - 1/32 * eps^5
        + 19/2048 * eps^7
        - 3/4096 * eps^9;
C1[2] = - 1/16 * eps^2
        + 1/32 * eps^4
        - 9/2048 * eps^6
        + 7/4096 * eps^8
        + 1/65536 * eps^10;
C1[3] = - 1/48 * eps^3
        + 3/256 * eps^5
        - 3/2048 * eps^7
        + 17/24576 * eps^9;
C1[4] = - 5/512 * eps^4
        + 3/512 * eps^6
        - 11/16384 * eps^8
        + 3/8192 * eps^10;
C1[5] = - 7/1280 * eps^5
        + 7/2048 * eps^7
        - 3/8192 * eps^9;
C1[6] = - 7/2048 * eps^6
        + 9/4096 * eps^8
        - 117/524288 * eps^10;
C1[7] = - 33/14336 * eps^7
        + 99/65536 * eps^9;
C1[8] = - 429/262144 * eps^8
        + 143/131072 * eps^10;
C1[9] = - 715/589824 * eps^9;
C1[10] = - 2431/2621440 * eps^10;
</pre></div><p>The function <em>tau</em>(<em>sigma</em>) = <em>s/</em>(<em>b</em> <em>A1</em>) = <em>sigma</em> + <em>B1</em>(<em>sigma</em>) may be inverted by series reversion giving</p>
<p>&nbsp; &nbsp; <em>sigma</em>(<em>tau</em>) = <em>tau</em> + sum<sub><em>j</em> = 1</sub> <em>C1'<sub>j</sub></em> sin(2 <em>j</em> <em>tau</em>)</p>
<p>where</p>
<div class="fragment"><pre class="fragment">
C1'[1] = + 1/2 * eps
         - 9/32 * eps^3
         + 205/1536 * eps^5
         - 4879/73728 * eps^7
         + 9039/327680 * eps^9;
C1'[2] = + 5/16 * eps^2
         - 37/96 * eps^4
         + 1335/4096 * eps^6
         - 86171/368640 * eps^8
         + 4119073/28311552 * eps^10;
C1'[3] = + 29/96 * eps^3
         - 75/128 * eps^5
         + 2901/4096 * eps^7
         - 443327/655360 * eps^9;
C1'[4] = + 539/1536 * eps^4
         - 2391/2560 * eps^6
         + 1082857/737280 * eps^8
         - 2722891/1548288 * eps^10;
C1'[5] = + 3467/7680 * eps^5
         - 28223/18432 * eps^7
         + 1361343/458752 * eps^9;
C1'[6] = + 38081/61440 * eps^6
         - 733437/286720 * eps^8
         + 10820079/1835008 * eps^10;
C1'[7] = + 459485/516096 * eps^7
         - 709743/163840 * eps^9;
C1'[8] = + 109167851/82575360 * eps^8
         - 550835669/74317824 * eps^10;
C1'[9] = + 83141299/41287680 * eps^9;
C1'[10] = + 9303339907/2972712960 * eps^10;
</pre></div><p>The reduced length is given by</p>
<p>&nbsp; &nbsp; <em>m/<em>b</em> =</em> sqrt(1 + <em>k<sup>2</sup></em> sin<sup>2</sup><em>sigma<sub>2</sub></em>) cos <em>sigma<sub>1</sub></em> sin <em>sigma<sub>2</sub></em> <br/>
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - sqrt(1 + <em>k<sup>2</sup></em> sin<sup>2</sup><em>sigma<sub>1</sub></em>) sin <em>sigma<sub>1</sub></em> cos <em>sigma<sub>2</sub></em> <br/>
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; - cos <em>sigma<sub>1</sub></em> cos <em>sigma<sub>2</sub></em> (<em>J</em>(<em>sigma<sub>2</sub></em>) - <em>J</em>(<em>sigma<sub>1</sub></em>))</p>
<p>where</p>
<p>&nbsp; &nbsp; <em>J</em>(<em>sigma</em>) = <em>I1</em>(<em>sigma</em>) - <em>I2</em>(<em>sigma</em>)<br/>
 &nbsp; &nbsp; <em>I2</em>(<em>sigma</em>) = <em>A2</em> (<em>sigma</em> + <em>B2</em>(<em>sigma</em>))<br/>
 &nbsp; &nbsp; <em>B2</em>(<em>sigma</em>) = sum<sub><em>j</em> = 1</sub> <em>C2<sub>j</sub></em> sin(2 <em>j</em> <em>sigma</em>)</p>
<div class="fragment"><pre class="fragment">
A2 = (1 + 1/4 * eps^2
        + 9/64 * eps^4
        + 25/256 * eps^6
        + 1225/16384 * eps^8
        + 3969/65536 * eps^10) * (1 - eps);
</pre></div><div class="fragment"><pre class="fragment">
C2[1] = + 1/2 * eps
        + 1/16 * eps^3
        + 1/32 * eps^5
        + 41/2048 * eps^7
        + 59/4096 * eps^9;
C2[2] = + 3/16 * eps^2
        + 1/32 * eps^4
        + 35/2048 * eps^6
        + 47/4096 * eps^8
        + 557/65536 * eps^10;
C2[3] = + 5/48 * eps^3
        + 5/256 * eps^5
        + 23/2048 * eps^7
        + 191/24576 * eps^9;
C2[4] = + 35/512 * eps^4
        + 7/512 * eps^6
        + 133/16384 * eps^8
        + 47/8192 * eps^10;
C2[5] = + 63/1280 * eps^5
        + 21/2048 * eps^7
        + 51/8192 * eps^9;
C2[6] = + 77/2048 * eps^6
        + 33/4096 * eps^8
        + 2607/524288 * eps^10;
C2[7] = + 429/14336 * eps^7
        + 429/65536 * eps^9;
C2[8] = + 6435/262144 * eps^8
        + 715/131072 * eps^10;
C2[9] = + 12155/589824 * eps^9;
C2[10] = + 46189/2621440 * eps^10;
</pre></div><p>The longitude is given in terms of the spherical longitude by</p>
<p>&nbsp; &nbsp; <em>lambda</em> = <em>omega</em> - <em>f</em> sin <em>alpha<sub>0</sub></em> <em>I3</em>(<em>sigma</em>)</p>
<p>where</p>
<p>&nbsp; &nbsp; <em>I3</em>(<em>sigma</em>) = <em>A3</em> (<em>sigma</em> + <em>B3</em>(<em>sigma</em>))<br/>
 &nbsp; &nbsp; <em>B3</em>(<em>sigma</em>) = sum<sub><em>j</em> = 1</sub> <em>C3<sub>j</sub></em> sin(2 <em>j</em> <em>sigma</em>)</p>
<p>and</p>
<div class="fragment"><pre class="fragment">
A3 = 1 - (1/2 - 1/2 * n) * eps
       - (1/4 + 1/8 * n - 3/8 * n^2) * eps^2
       - (1/16 + 3/16 * n + 1/16 * n^2 - 5/16 * n^3) * eps^3
       - (3/64 + 1/32 * n + 5/32 * n^2 + 5/128 * n^3 - 35/128 * n^4) * eps^4
       - (3/128 + 5/128 * n + 5/256 * n^2 + 35/256 * n^3 + 7/256 * n^4) * eps^5
       - (5/256 + 15/1024 * n + 35/1024 * n^2 + 7/512 * n^3) * eps^6
       - (25/2048 + 35/2048 * n + 21/2048 * n^2) * eps^7
       - (175/16384 + 35/4096 * n) * eps^8
       - 245/32768 * eps^9;
</pre></div><div class="fragment"><pre class="fragment">
C3[1] = + (1/4 - 1/4 * n) * eps
        + (1/8 - 1/8 * n^2) * eps^2
        + (3/64 + 3/64 * n - 1/64 * n^2 - 5/64 * n^3) * eps^3
        + (5/128 + 1/64 * n + 1/64 * n^2 - 1/64 * n^3 - 7/128 * n^4) * eps^4
        + (3/128 + 11/512 * n + 3/512 * n^2 + 1/256 * n^3 - 7/512 * n^4) * eps^5
        + (21/1024 + 5/512 * n + 13/1024 * n^2 + 1/512 * n^3) * eps^6
        + (243/16384 + 189/16384 * n + 83/16384 * n^2) * eps^7
        + (435/32768 + 109/16384 * n) * eps^8
        + 345/32768 * eps^9;
C3[2] = + (1/16 - 3/32 * n + 1/32 * n^2) * eps^2
        + (3/64 - 1/32 * n - 3/64 * n^2 + 1/32 * n^3) * eps^3
        + (3/128 + 1/128 * n - 9/256 * n^2 - 3/128 * n^3 + 7/256 * n^4) * eps^4
        + (5/256 + 1/256 * n - 1/128 * n^2 - 7/256 * n^3 - 3/256 * n^4) * eps^5
        + (27/2048 + 69/8192 * n - 39/8192 * n^2 - 47/4096 * n^3) * eps^6
        + (187/16384 + 39/8192 * n + 31/16384 * n^2) * eps^7
        + (287/32768 + 47/8192 * n) * eps^8
        + 255/32768 * eps^9;
C3[3] = + (5/192 - 3/64 * n + 5/192 * n^2 - 1/192 * n^3) * eps^3
        + (3/128 - 5/192 * n - 1/64 * n^2 + 5/192 * n^3 - 1/128 * n^4) * eps^4
        + (7/512 - 1/384 * n - 77/3072 * n^2 + 5/3072 * n^3 + 65/3072 * n^4) * eps^5
        + (3/256 - 1/1024 * n - 71/6144 * n^2 - 47/3072 * n^3) * eps^6
        + (139/16384 + 143/49152 * n - 383/49152 * n^2) * eps^7
        + (243/32768 + 95/49152 * n) * eps^8
        + 581/98304 * eps^9;
C3[4] = + (7/512 - 7/256 * n + 5/256 * n^2 - 7/1024 * n^3 + 1/1024 * n^4) * eps^4
        + (7/512 - 5/256 * n - 7/2048 * n^2 + 9/512 * n^3 - 21/2048 * n^4) * eps^5
        + (9/1024 - 43/8192 * n - 129/8192 * n^2 + 39/4096 * n^3) * eps^6
        + (127/16384 - 23/8192 * n - 165/16384 * n^2) * eps^7
        + (193/32768 + 3/8192 * n) * eps^8
        + 171/32768 * eps^9;
C3[5] = + (21/2560 - 9/512 * n + 15/1024 * n^2 - 7/1024 * n^3 + 9/5120 * n^4) * eps^5
        + (9/1024 - 15/1024 * n + 3/2048 * n^2 + 57/5120 * n^3) * eps^6
        + (99/16384 - 91/16384 * n - 781/81920 * n^2) * eps^7
        + (179/32768 - 55/16384 * n) * eps^8
        + 141/32768 * eps^9;
C3[6] = + (11/2048 - 99/8192 * n + 275/24576 * n^2 - 77/12288 * n^3) * eps^6
        + (99/16384 - 275/24576 * n + 55/16384 * n^2) * eps^7
        + (143/32768 - 253/49152 * n) * eps^8
        + 33/8192 * eps^9;
C3[7] = + (429/114688 - 143/16384 * n + 143/16384 * n^2) * eps^7
        + (143/32768 - 143/16384 * n) * eps^8
        + 429/131072 * eps^9;
C3[8] = + (715/262144 - 429/65536 * n) * eps^8
        + 429/131072 * eps^9;
C3[9] = + 2431/1179648 * eps^9;
</pre></div><p>The formula for area between the geodesic and the equator is given in Sec. 15 of <a href="http://arxiv.org/abs/1102.1215">arxiv:1102.1215</a> in terms of <em>S</em>,</p>
<p>&nbsp; &nbsp; <em>S</em> = <em>c<sup>2</sup></em> <em>alpha</em> + <em>e<sup>2</sup></em> <em>a<sup>2</sup></em> cos <em>alpha<sub>0</sub></em> sin <em>alpha<sub>0</sub></em> <em>I4</em>(<em>sigma</em>)</p>
<p>where</p>
<p>&nbsp; &nbsp; <em>I4</em>(<em>sigma</em>) = sum<sub><em>j</em> = 0</sub> <em>C4<sub>j</sub></em> cos((2<em>j</em> + 1) <em>sigma</em>)</p>
<p>with</p>
<div class="fragment"><pre class="fragment">
C4[0] = + (2/3 - 1/15 * ep2 + 4/105 * ep2^2 - 8/315 * ep2^3 + 64/3465 * ep2^4 - 128/9009 * ep2^5 + 512/45045 * ep2^6 - 1024/109395 * ep2^7 + 16384/2078505 * ep2^8 - 32768/4849845 * ep2^9)
        - (1/20 - 1/35 * ep2 + 2/105 * ep2^2 - 16/1155 * ep2^3 + 32/3003 * ep2^4 - 128/15015 * ep2^5 + 256/36465 * ep2^6 - 4096/692835 * ep2^7 + 8192/1616615 * ep2^8) * k2
        + (1/42 - 1/63 * ep2 + 8/693 * ep2^2 - 80/9009 * ep2^3 + 64/9009 * ep2^4 - 128/21879 * ep2^5 + 2048/415701 * ep2^6 - 4096/969969 * ep2^7) * k2^2
        - (1/72 - 1/99 * ep2 + 10/1287 * ep2^2 - 8/1287 * ep2^3 + 112/21879 * ep2^4 - 1792/415701 * ep2^5 + 512/138567 * ep2^6) * k2^3
        + (1/110 - 1/143 * ep2 + 4/715 * ep2^2 - 56/12155 * ep2^3 + 896/230945 * ep2^4 - 768/230945 * ep2^5) * k2^4
        - (1/156 - 1/195 * ep2 + 14/3315 * ep2^2 - 224/62985 * ep2^3 + 64/20995 * ep2^4) * k2^5
        + (1/210 - 1/255 * ep2 + 16/4845 * ep2^2 - 32/11305 * ep2^3) * k2^6
        - (1/272 - 1/323 * ep2 + 6/2261 * ep2^2) * k2^7
        + (1/342 - 1/399 * ep2) * k2^8
        - 1/420 * k2^9;
C4[1] = + (1/180 - 1/315 * ep2 + 2/945 * ep2^2 - 16/10395 * ep2^3 + 32/27027 * ep2^4 - 128/135135 * ep2^5 + 256/328185 * ep2^6 - 4096/6235515 * ep2^7 + 8192/14549535 * ep2^8) * k2
        - (1/252 - 1/378 * ep2 + 4/2079 * ep2^2 - 40/27027 * ep2^3 + 32/27027 * ep2^4 - 64/65637 * ep2^5 + 1024/1247103 * ep2^6 - 2048/2909907 * ep2^7) * k2^2
        + (1/360 - 1/495 * ep2 + 2/1287 * ep2^2 - 8/6435 * ep2^3 + 112/109395 * ep2^4 - 1792/2078505 * ep2^5 + 512/692835 * ep2^6) * k2^3
        - (1/495 - 2/1287 * ep2 + 8/6435 * ep2^2 - 112/109395 * ep2^3 + 1792/2078505 * ep2^4 - 512/692835 * ep2^5) * k2^4
        + (5/3276 - 1/819 * ep2 + 2/1989 * ep2^2 - 32/37791 * ep2^3 + 64/88179 * ep2^4) * k2^5
        - (1/840 - 1/1020 * ep2 + 4/4845 * ep2^2 - 8/11305 * ep2^3) * k2^6
        + (7/7344 - 7/8721 * ep2 + 2/2907 * ep2^2) * k2^7
        - (2/2565 - 4/5985 * ep2) * k2^8
        + 1/1540 * k2^9;
C4[2] = + (1/2100 - 1/3150 * ep2 + 4/17325 * ep2^2 - 8/45045 * ep2^3 + 32/225225 * ep2^4 - 64/546975 * ep2^5 + 1024/10392525 * ep2^6 - 2048/24249225 * ep2^7) * k2^2
        - (1/1800 - 1/2475 * ep2 + 2/6435 * ep2^2 - 8/32175 * ep2^3 + 112/546975 * ep2^4 - 1792/10392525 * ep2^5 + 512/3464175 * ep2^6) * k2^3
        + (1/1925 - 2/5005 * ep2 + 8/25025 * ep2^2 - 16/60775 * ep2^3 + 256/1154725 * ep2^4 - 1536/8083075 * ep2^5) * k2^4
        - (1/2184 - 1/2730 * ep2 + 1/3315 * ep2^2 - 16/62985 * ep2^3 + 32/146965 * ep2^4) * k2^5
        + (1/2520 - 1/3060 * ep2 + 4/14535 * ep2^2 - 8/33915 * ep2^3) * k2^6
        - (7/20400 - 7/24225 * ep2 + 2/8075 * ep2^2) * k2^7
        + (14/47025 - 4/15675 * ep2) * k2^8
        - 1/3850 * k2^9;
C4[3] = + (1/17640 - 1/24255 * ep2 + 2/63063 * ep2^2 - 8/315315 * ep2^3 + 16/765765 * ep2^4 - 256/14549535 * ep2^5 + 512/33948915 * ep2^6) * k2^3
        - (1/10780 - 1/14014 * ep2 + 2/35035 * ep2^2 - 4/85085 * ep2^3 + 64/1616615 * ep2^4 - 384/11316305 * ep2^5) * k2^4
        + (5/45864 - 1/11466 * ep2 + 1/13923 * ep2^2 - 16/264537 * ep2^3 + 32/617253 * ep2^4) * k2^5
        - (1/8820 - 1/10710 * ep2 + 8/101745 * ep2^2 - 16/237405 * ep2^3) * k2^6
        + (1/8976 - 1/10659 * ep2 + 2/24871 * ep2^2) * k2^7
        - (1/9405 - 2/21945 * ep2) * k2^8
        + 1/10010 * k2^9;
C4[4] = + (1/124740 - 1/162162 * ep2 + 2/405405 * ep2^2 - 4/984555 * ep2^3 + 64/18706545 * ep2^4 - 128/43648605 * ep2^5) * k2^4
        - (1/58968 - 1/73710 * ep2 + 1/89505 * ep2^2 - 16/1700595 * ep2^3 + 32/3968055 * ep2^4) * k2^5
        + (1/41580 - 1/50490 * ep2 + 8/479655 * ep2^2 - 16/1119195 * ep2^3) * k2^6
        - (7/242352 - 7/287793 * ep2 + 2/95931 * ep2^2) * k2^7
        + (7/220077 - 2/73359 * ep2) * k2^8
        - 1/30030 * k2^9;
C4[5] = + (1/792792 - 1/990990 * ep2 + 1/1203345 * ep2^2 - 16/22863555 * ep2^3 + 32/53348295 * ep2^4) * k2^5
        - (1/304920 - 1/370260 * ep2 + 4/1758735 * ep2^2 - 8/4103715 * ep2^3) * k2^6
        + (7/1283568 - 7/1524237 * ep2 + 2/508079 * ep2^2) * k2^7
        - (2/268983 - 4/627627 * ep2) * k2^8
        + 1/110110 * k2^9;
C4[6] = + (1/4684680 - 1/5688540 * ep2 + 4/27020565 * ep2^2 - 8/63047985 * ep2^3) * k2^6
        - (1/1516944 - 1/1801371 * ep2 + 2/4203199 * ep2^2) * k2^7
        + (2/1589445 - 4/3708705 * ep2) * k2^8
        - 1/520520 * k2^9;
C4[7] = + (1/26254800 - 1/31177575 * ep2 + 2/72747675 * ep2^2) * k2^7
        - (1/7335900 - 1/8558550 * ep2) * k2^8
        + 1/3403400 * k2^9;
C4[8] = + (1/141338340 - 1/164894730 * ep2) * k2^8
        - 1/34714680 * k2^9;
C4[9] = + 1/737176440 * k2^9;
</pre></div><center> Back to <a class="el" href="transversemercator.html">Transverse Mercator Projection</a>. Forward to <a class="el" href="geocentric.html">Geocentric coordinates</a>. Up to <a class="el" href="index.html#contents">Contents</a>. </center> </div>
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