/usr/share/doc/geographiclib/html/auxlat.html is in geographiclib-doc 1.49-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 | <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=9"/>
<meta name="generator" content="Doxygen 1.8.13"/>
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<title>GeographicLib: Auxiliary latitudes</title>
<link href="tabs.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="jquery.js"></script>
<script type="text/javascript" src="dynsections.js"></script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
extensions: ["tex2jax.js"],
jax: ["input/TeX","output/HTML-CSS"],
});
</script><script type="text/javascript" src="/usr/share/javascript/mathjax/MathJax.js/MathJax.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
</head>
<body>
<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
<div id="titlearea">
<table cellspacing="0" cellpadding="0">
<tbody>
<tr style="height: 56px;">
<td id="projectalign" style="padding-left: 0.5em;">
<div id="projectname">GeographicLib
 <span id="projectnumber">1.49</span>
</div>
</td>
</tr>
</tbody>
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.8.13 -->
<script type="text/javascript" src="menudata.js"></script>
<script type="text/javascript" src="menu.js"></script>
<script type="text/javascript">
$(function() {
initMenu('',false,false,'search.php','Search');
});
</script>
<div id="main-nav"></div>
</div><!-- top -->
<div class="header">
<div class="headertitle">
<div class="title">Auxiliary latitudes </div> </div>
</div><!--header-->
<div class="contents">
<div class="textblock"><center> Back to <a class="el" href="geocentric.html">Geocentric coordinates</a>. Forward to <a class="el" href="highprec.html">Support for high precision arithmetic</a>. Up to <a class="el" href="index.html#contents">Contents</a>. </center><p>Go to</p><ul>
<li><a class="el" href="auxlat.html#auxlatformula">Series approximations for conversions</a></li>
<li><a class="el" href="auxlat.html#auxlattable">Series approximations in tabular form</a></li>
<li><a class="el" href="auxlat.html#auxlaterror">Truncation errors</a></li>
</ul>
<p>Six latitudes are used by <a class="el" href="namespaceGeographicLib.html" title="Namespace for GeographicLib. ">GeographicLib</a>:</p><ul>
<li>φ, the (geographic) latitude;</li>
<li>β, the parametric latitude;</li>
<li>θ, the geocentric latitude;</li>
<li>μ, the rectifying latitude;</li>
<li>χ, the conformal latitude;</li>
<li>ξ, the authalic latitude.</li>
</ul>
<p>The last five of these are called <em>auxiliary latitudes</em>. These quantities are all defined in the <a href="https://en.wikipedia.org/wiki/Latitude#Auxiliary_latitudes">Wikipedia article on latitudes</a>.</p>
<p>In addition there's the isometric latitude, ψ, defined by ψ = gd<sup>−1</sup>χ = sinh<sup>−1</sup> tanχ and χ = gdψ = tan<sup>−1</sup> sinhψ. This is not an angle-like variable (for example, it diverges at the poles) and so we don't treat it further here. However conversions between ψ and any of the auxiliary latitudes is easily accomplished via an intermediate conversion to χ.</p>
<p>The relations between φ, β, and θ are all simple elementary functions. The latitudes χ and ξ can be expressed as elementary functions of φ; however, these functions can only be inverted iteratively. The rectifying latitude μ as a function of φ (or β) involves the incomplete elliptic integral of the second kind (which is not an elementary function) and this needs to be inverted iteratively. The <a class="el" href="classGeographicLib_1_1Ellipsoid.html" title="Properties of an ellipsoid. ">Ellipsoid</a> class evaluates all the auxiliary latitudes (and the corresponding inverse relations) in terms of their basic definitions.</p>
<p>An alternative method of evaluating these auxiliary latitudes is in terms of trigonometric series. This offers some advantages:</p><ul>
<li>these series give a uniform way of expressing any latitude in terms of any other latitude;</li>
<li>the evaluation may be faster, particularly if <a href="https://en.wikipedia.org/wiki/Clenshaw_algorithm#Meridian_arc_length_on_the_ellipsoid">Clenshaw summation</a> is used;</li>
<li>provided that the flattening is sufficiently small, the result may be more accurate.</li>
</ul>
<p>Here we give the complete matrix of relations between all six latitudes; there are 30 (= 6 × 5) such relations. These expansions complement the work of</p><ul>
<li>O. S. Adams, <a href="https://docs.lib.noaa.gov/rescue/cgs_specpubs/QB275U35no671921.pdf">Latitude developments connected with geodesy and cartography</a>, Spec. Pub. 67 (US Coast and Geodetic Survey, 1921).</li>
<li>P. Osborne, <a href="https://dx.doi.org/10.5281/zenodo.35392">The Mercator Projections</a> (2013), Chap. 5.</li>
<li>S. Orihuela, <a href="https://sites.google.com/site/geodesiafich/funciones_latitud.pdf">Funciones de Latitud</a> (2013).</li>
</ul>
<p>Here, the expansions are in terms of the third flattening <em>n</em> = (<em>a</em> − <em>b</em>)/(<em>a</em> + <em>b</em>). This choice of expansion parameter results in expansions in which half the coefficients vanish for all relations between φ, β, θ, and μ. In addition, the expansions converge for <em>b</em>/<em>a</em> ∈ (0, ∞). These expansions were obtained with the the maxima code, <a href="auxlat.mac">auxlat.mac</a>.</p>
<p>Adams (1921) uses the eccentricity squared <em>e</em><sup>2</sup> as the expansion parameter, but the resulting series only converge for <em>b</em>/<em>a</em> ∈ (0, √2). In addition, it is shown in <a class="el" href="auxlat.html#auxlaterror">Truncation errors</a>, that the errors when the series are truncated are much worse than for the corresponding series in <em>n</em>.</p>
<h1><a class="anchor" id="auxlatformula"></a>
Series approximations for conversions</h1>
<p>Here are the relations between φ, β, θ, and μ carried out to 4th order in <em>n</em>: </p><p class="formulaDsp">
\[ \begin{align} \beta-\phi&=\textstyle{} -n\sin 2\phi +\frac{1}{2}n^{2}\sin 4\phi -\frac{1}{3}n^{3}\sin 6\phi +\frac{1}{4}n^{4}\sin 8\phi -\ldots\\ \phi-\beta&=\textstyle{} +n\sin 2\beta +\frac{1}{2}n^{2}\sin 4\beta +\frac{1}{3}n^{3}\sin 6\beta +\frac{1}{4}n^{4}\sin 8\beta +\ldots\\ \theta-\phi&=\textstyle{} -\bigl(2n-2n^{3}\bigr)\sin 2\phi +\bigl(2n^{2}-4n^{4}\bigr)\sin 4\phi -\frac{8}{3}n^{3}\sin 6\phi +4n^{4}\sin 8\phi -\ldots\\ \phi-\theta&=\textstyle{} +\bigl(2n-2n^{3}\bigr)\sin 2\theta +\bigl(2n^{2}-4n^{4}\bigr)\sin 4\theta +\frac{8}{3}n^{3}\sin 6\theta +4n^{4}\sin 8\theta +\ldots\\ \theta-\beta&=\textstyle{} -n\sin 2\beta +\frac{1}{2}n^{2}\sin 4\beta -\frac{1}{3}n^{3}\sin 6\beta +\frac{1}{4}n^{4}\sin 8\beta -\ldots\\ \beta-\theta&=\textstyle{} +n\sin 2\theta +\frac{1}{2}n^{2}\sin 4\theta +\frac{1}{3}n^{3}\sin 6\theta +\frac{1}{4}n^{4}\sin 8\theta +\ldots\\ \mu-\phi&=\textstyle{} -\bigl(\frac{3}{2}n-\frac{9}{16}n^{3}\bigr)\sin 2\phi +\bigl(\frac{15}{16}n^{2}-\frac{15}{32}n^{4}\bigr)\sin 4\phi -\frac{35}{48}n^{3}\sin 6\phi +\frac{315}{512}n^{4}\sin 8\phi -\ldots\\ \phi-\mu&=\textstyle{} +\bigl(\frac{3}{2}n-\frac{27}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{21}{16}n^{2}-\frac{55}{32}n^{4}\bigr)\sin 4\mu +\frac{151}{96}n^{3}\sin 6\mu +\frac{1097}{512}n^{4}\sin 8\mu +\ldots\\ \mu-\beta&=\textstyle{} -\bigl(\frac{1}{2}n-\frac{3}{16}n^{3}\bigr)\sin 2\beta -\bigl(\frac{1}{16}n^{2}-\frac{1}{32}n^{4}\bigr)\sin 4\beta -\frac{1}{48}n^{3}\sin 6\beta -\frac{5}{512}n^{4}\sin 8\beta -\ldots\\ \beta-\mu&=\textstyle{} +\bigl(\frac{1}{2}n-\frac{9}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{5}{16}n^{2}-\frac{37}{96}n^{4}\bigr)\sin 4\mu +\frac{29}{96}n^{3}\sin 6\mu +\frac{539}{1536}n^{4}\sin 8\mu +\ldots\\ \mu-\theta&=\textstyle{} +\bigl(\frac{1}{2}n+\frac{13}{16}n^{3}\bigr)\sin 2\theta -\bigl(\frac{1}{16}n^{2}-\frac{33}{32}n^{4}\bigr)\sin 4\theta -\frac{5}{16}n^{3}\sin 6\theta -\frac{261}{512}n^{4}\sin 8\theta -\ldots\\ \theta-\mu&=\textstyle{} -\bigl(\frac{1}{2}n+\frac{23}{32}n^{3}\bigr)\sin 2\mu +\bigl(\frac{5}{16}n^{2}-\frac{5}{96}n^{4}\bigr)\sin 4\mu +\frac{1}{32}n^{3}\sin 6\mu +\frac{283}{1536}n^{4}\sin 8\mu +\ldots\\ \end{align} \]
</p>
<p>Here are the remaining relations (including χ and ξ) carried out to 3rd order in <em>n</em>: </p><p class="formulaDsp">
\[ \begin{align} \chi-\phi&=\textstyle{} -\bigl(2n-\frac{2}{3}n^{2}-\frac{4}{3}n^{3}\bigr)\sin 2\phi +\bigl(\frac{5}{3}n^{2}-\frac{16}{15}n^{3}\bigr)\sin 4\phi -\frac{26}{15}n^{3}\sin 6\phi +\ldots\\ \phi-\chi&=\textstyle{} +\bigl(2n-\frac{2}{3}n^{2}-2n^{3}\bigr)\sin 2\chi +\bigl(\frac{7}{3}n^{2}-\frac{8}{5}n^{3}\bigr)\sin 4\chi +\frac{56}{15}n^{3}\sin 6\chi +\ldots\\ \chi-\beta&=\textstyle{} -\bigl(n-\frac{2}{3}n^{2}\bigr)\sin 2\beta +\bigl(\frac{1}{6}n^{2}-\frac{2}{5}n^{3}\bigr)\sin 4\beta -\frac{1}{15}n^{3}\sin 6\beta +\ldots\\ \beta-\chi&=\textstyle{} +\bigl(n-\frac{2}{3}n^{2}-\frac{1}{3}n^{3}\bigr)\sin 2\chi +\bigl(\frac{5}{6}n^{2}-\frac{14}{15}n^{3}\bigr)\sin 4\chi +\frac{16}{15}n^{3}\sin 6\chi +\ldots\\ \chi-\theta&=\textstyle{} +\bigl(\frac{2}{3}n^{2}+\frac{2}{3}n^{3}\bigr)\sin 2\theta -\bigl(\frac{1}{3}n^{2}-\frac{4}{15}n^{3}\bigr)\sin 4\theta -\frac{2}{5}n^{3}\sin 6\theta -\ldots\\ \theta-\chi&=\textstyle{} -\bigl(\frac{2}{3}n^{2}+\frac{2}{3}n^{3}\bigr)\sin 2\chi +\bigl(\frac{1}{3}n^{2}-\frac{4}{15}n^{3}\bigr)\sin 4\chi +\frac{2}{5}n^{3}\sin 6\chi +\ldots\\ \chi-\mu&=\textstyle{} -\bigl(\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{37}{96}n^{3}\bigr)\sin 2\mu -\bigl(\frac{1}{48}n^{2}+\frac{1}{15}n^{3}\bigr)\sin 4\mu -\frac{17}{480}n^{3}\sin 6\mu -\ldots\\ \mu-\chi&=\textstyle{} +\bigl(\frac{1}{2}n-\frac{2}{3}n^{2}+\frac{5}{16}n^{3}\bigr)\sin 2\chi +\bigl(\frac{13}{48}n^{2}-\frac{3}{5}n^{3}\bigr)\sin 4\chi +\frac{61}{240}n^{3}\sin 6\chi +\ldots\\ \xi-\phi&=\textstyle{} -\bigl(\frac{4}{3}n+\frac{4}{45}n^{2}-\frac{88}{315}n^{3}\bigr)\sin 2\phi +\bigl(\frac{34}{45}n^{2}+\frac{8}{105}n^{3}\bigr)\sin 4\phi -\frac{1532}{2835}n^{3}\sin 6\phi +\ldots\\ \phi-\xi&=\textstyle{} +\bigl(\frac{4}{3}n+\frac{4}{45}n^{2}-\frac{16}{35}n^{3}\bigr)\sin 2\xi +\bigl(\frac{46}{45}n^{2}+\frac{152}{945}n^{3}\bigr)\sin 4\xi +\frac{3044}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\beta&=\textstyle{} -\bigl(\frac{1}{3}n+\frac{4}{45}n^{2}-\frac{32}{315}n^{3}\bigr)\sin 2\beta -\bigl(\frac{7}{90}n^{2}+\frac{4}{315}n^{3}\bigr)\sin 4\beta -\frac{83}{2835}n^{3}\sin 6\beta -\ldots\\ \beta-\xi&=\textstyle{} +\bigl(\frac{1}{3}n+\frac{4}{45}n^{2}-\frac{46}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{17}{90}n^{2}+\frac{68}{945}n^{3}\bigr)\sin 4\xi +\frac{461}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\theta&=\textstyle{} +\bigl(\frac{2}{3}n-\frac{4}{45}n^{2}+\frac{62}{105}n^{3}\bigr)\sin 2\theta +\bigl(\frac{4}{45}n^{2}-\frac{32}{315}n^{3}\bigr)\sin 4\theta -\frac{524}{2835}n^{3}\sin 6\theta -\ldots\\ \theta-\xi&=\textstyle{} -\bigl(\frac{2}{3}n-\frac{4}{45}n^{2}+\frac{158}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{16}{45}n^{2}-\frac{16}{945}n^{3}\bigr)\sin 4\xi -\frac{232}{2835}n^{3}\sin 6\xi +\ldots\\ \xi-\mu&=\textstyle{} +\bigl(\frac{1}{6}n-\frac{4}{45}n^{2}-\frac{817}{10080}n^{3}\bigr)\sin 2\mu +\bigl(\frac{49}{720}n^{2}-\frac{2}{35}n^{3}\bigr)\sin 4\mu +\frac{4463}{90720}n^{3}\sin 6\mu +\ldots\\ \mu-\xi&=\textstyle{} -\bigl(\frac{1}{6}n-\frac{4}{45}n^{2}-\frac{121}{1680}n^{3}\bigr)\sin 2\xi -\bigl(\frac{29}{720}n^{2}-\frac{26}{945}n^{3}\bigr)\sin 4\xi -\frac{1003}{45360}n^{3}\sin 6\xi -\ldots\\ \xi-\chi&=\textstyle{} +\bigl(\frac{2}{3}n-\frac{34}{45}n^{2}+\frac{46}{315}n^{3}\bigr)\sin 2\chi +\bigl(\frac{19}{45}n^{2}-\frac{256}{315}n^{3}\bigr)\sin 4\chi +\frac{248}{567}n^{3}\sin 6\chi +\ldots\\ \chi-\xi&=\textstyle{} -\bigl(\frac{2}{3}n-\frac{34}{45}n^{2}+\frac{88}{315}n^{3}\bigr)\sin 2\xi +\bigl(\frac{1}{45}n^{2}-\frac{184}{945}n^{3}\bigr)\sin 4\xi -\frac{106}{2835}n^{3}\sin 6\xi -\ldots\\ \end{align} \]
</p>
<h1><a class="anchor" id="auxlattable"></a>
Series approximations in tabular form</h1>
<p>Finally, this is a listing of all the coefficients for the expansions carried out to 8th order in <em>n</em>. Here's how to interpret this data: the 5th line for φ − θ is <code>[32/5, 0, -32, 0]</code>; this means that the coefficient of sin(10θ) is [(32/5)<em>n</em><sup>5</sup> − 32<em>n</em><sup>7</sup> + <em>O</em>(<em>n</em><sup>9</sup>)]. </p>
<p>β − φ:<br />
<code><small>    [-1, 0, 0, 0, 0, 0, 0, 0]<br />
   [1/2, 0, 0, 0, 0, 0, 0]<br />
   [-1/3, 0, 0, 0, 0, 0]<br />
   [1/4, 0, 0, 0, 0]<br />
   [-1/5, 0, 0, 0]<br />
   [1/6, 0, 0]<br />
   [-1/7, 0]<br />
   [1/8]<br />
</small></code> </p>
<p>φ − β:<br />
<code><small>    [1, 0, 0, 0, 0, 0, 0, 0]<br />
   [1/2, 0, 0, 0, 0, 0, 0]<br />
   [1/3, 0, 0, 0, 0, 0]<br />
   [1/4, 0, 0, 0, 0]<br />
   [1/5, 0, 0, 0]<br />
   [1/6, 0, 0]<br />
   [1/7, 0]<br />
   [1/8]<br />
</small></code> </p>
<p>θ − φ:<br />
<code><small>    [-2, 0, 2, 0, -2, 0, 2, 0]<br />
   [2, 0, -4, 0, 6, 0, -8]<br />
   [-8/3, 0, 8, 0, -16, 0]<br />
   [4, 0, -16, 0, 40]<br />
   [-32/5, 0, 32, 0]<br />
   [32/3, 0, -64]<br />
   [-128/7, 0]<br />
   [32]<br />
</small></code> </p>
<p>φ − θ:<br />
<code><small>    [2, 0, -2, 0, 2, 0, -2, 0]<br />
   [2, 0, -4, 0, 6, 0, -8]<br />
   [8/3, 0, -8, 0, 16, 0]<br />
   [4, 0, -16, 0, 40]<br />
   [32/5, 0, -32, 0]<br />
   [32/3, 0, -64]<br />
   [128/7, 0]<br />
   [32]<br />
</small></code> </p>
<p>θ − β:<br />
<code><small>    [-1, 0, 0, 0, 0, 0, 0, 0]<br />
   [1/2, 0, 0, 0, 0, 0, 0]<br />
   [-1/3, 0, 0, 0, 0, 0]<br />
   [1/4, 0, 0, 0, 0]<br />
   [-1/5, 0, 0, 0]<br />
   [1/6, 0, 0]<br />
   [-1/7, 0]<br />
   [1/8]<br />
</small></code> </p>
<p>β − θ:<br />
<code><small>    [1, 0, 0, 0, 0, 0, 0, 0]<br />
   [1/2, 0, 0, 0, 0, 0, 0]<br />
   [1/3, 0, 0, 0, 0, 0]<br />
   [1/4, 0, 0, 0, 0]<br />
   [1/5, 0, 0, 0]<br />
   [1/6, 0, 0]<br />
   [1/7, 0]<br />
   [1/8]<br />
</small></code> </p>
<p>μ − φ:<br />
<code><small>    [-3/2, 0, 9/16, 0, -3/32, 0, 57/2048, 0]<br />
   [15/16, 0, -15/32, 0, 135/2048, 0, -105/4096]<br />
   [-35/48, 0, 105/256, 0, -105/2048, 0]<br />
   [315/512, 0, -189/512, 0, 693/16384]<br />
   [-693/1280, 0, 693/2048, 0]<br />
   [1001/2048, 0, -1287/4096]<br />
   [-6435/14336, 0]<br />
   [109395/262144]<br />
</small></code> </p>
<p>φ − μ:<br />
<code><small>    [3/2, 0, -27/32, 0, 269/512, 0, -6607/24576, 0]<br />
   [21/16, 0, -55/32, 0, 6759/4096, 0, -155113/122880]<br />
   [151/96, 0, -417/128, 0, 87963/20480, 0]<br />
   [1097/512, 0, -15543/2560, 0, 2514467/245760]<br />
   [8011/2560, 0, -69119/6144, 0]<br />
   [293393/61440, 0, -5962461/286720]<br />
   [6459601/860160, 0]<br />
   [332287993/27525120]<br />
</small></code> </p>
<p>μ − β:<br />
<code><small>    [-1/2, 0, 3/16, 0, -1/32, 0, 19/2048, 0]<br />
   [-1/16, 0, 1/32, 0, -9/2048, 0, 7/4096]<br />
   [-1/48, 0, 3/256, 0, -3/2048, 0]<br />
   [-5/512, 0, 3/512, 0, -11/16384]<br />
   [-7/1280, 0, 7/2048, 0]<br />
   [-7/2048, 0, 9/4096]<br />
   [-33/14336, 0]<br />
   [-429/262144]<br />
</small></code> </p>
<p>β − μ:<br />
<code><small>    [1/2, 0, -9/32, 0, 205/1536, 0, -4879/73728, 0]<br />
   [5/16, 0, -37/96, 0, 1335/4096, 0, -86171/368640]<br />
   [29/96, 0, -75/128, 0, 2901/4096, 0]<br />
   [539/1536, 0, -2391/2560, 0, 1082857/737280]<br />
   [3467/7680, 0, -28223/18432, 0]<br />
   [38081/61440, 0, -733437/286720]<br />
   [459485/516096, 0]<br />
   [109167851/82575360]<br />
</small></code> </p>
<p>μ − θ:<br />
<code><small>    [1/2, 0, 13/16, 0, -15/32, 0, 509/2048, 0]<br />
   [-1/16, 0, 33/32, 0, -1673/2048, 0, 2599/4096]<br />
   [-5/16, 0, 349/256, 0, -2989/2048, 0]<br />
   [-261/512, 0, 963/512, 0, -43531/16384]<br />
   [-921/1280, 0, 5545/2048, 0]<br />
   [-6037/6144, 0, 16617/4096]<br />
   [-19279/14336, 0]<br />
   [-490925/262144]<br />
</small></code> </p>
<p>θ − μ:<br />
<code><small>    [-1/2, 0, -23/32, 0, 499/1536, 0, -14321/73728, 0]<br />
   [5/16, 0, -5/96, 0, 6565/12288, 0, -201467/368640]<br />
   [1/32, 0, -77/128, 0, 2939/4096, 0]<br />
   [283/1536, 0, -4037/7680, 0, 1155049/737280]<br />
   [1301/7680, 0, -19465/18432, 0]<br />
   [17089/61440, 0, -442269/286720]<br />
   [198115/516096, 0]<br />
   [48689387/82575360]<br />
</small></code> </p>
<p>χ − φ:<br />
<code><small>    [-2, 2/3, 4/3, -82/45, 32/45, 4642/4725, -8384/4725, 1514/1323]<br />
   [5/3, -16/15, -13/9, 904/315, -1522/945, -2288/1575, 142607/42525]<br />
   [-26/15, 34/21, 8/5, -12686/2835, 44644/14175, 120202/51975]<br />
   [1237/630, -12/5, -24832/14175, 1077964/155925, -1097407/187110]<br />
   [-734/315, 109598/31185, 1040/567, -12870194/1216215]<br />
   [444337/155925, -941912/184275, -126463/72765]<br />
   [-2405834/675675, 3463678/467775]<br />
   [256663081/56756700]<br />
</small></code> </p>
<p>φ − χ:<br />
<code><small>    [2, -2/3, -2, 116/45, 26/45, -2854/675, 16822/4725, 189416/99225]<br />
   [7/3, -8/5, -227/45, 2704/315, 2323/945, -31256/1575, 141514/8505]<br />
   [56/15, -136/35, -1262/105, 73814/2835, 98738/14175, -2363828/31185]<br />
   [4279/630, -332/35, -399572/14175, 11763988/155925, 14416399/935550]<br />
   [4174/315, -144838/6237, -2046082/31185, 258316372/1216215]<br />
   [601676/22275, -115444544/2027025, -2155215124/14189175]<br />
   [38341552/675675, -170079376/1216215]<br />
   [1383243703/11351340]<br />
</small></code> </p>
<p>χ − β:<br />
<code><small>    [-1, 2/3, 0, -16/45, 2/5, -998/4725, -34/4725, 1384/11025]<br />
   [1/6, -2/5, 19/45, -22/105, -2/27, 1268/4725, -12616/42525]<br />
   [-1/15, 16/105, -22/105, 116/567, -1858/14175, 1724/51975]<br />
   [17/1260, -8/105, 2123/14175, -26836/155925, 115249/935550]<br />
   [-1/105, 128/4455, -424/6237, 140836/1216215]<br />
   [149/311850, -31232/2027025, 210152/4729725]<br />
   [-499/225225, 30208/6081075]<br />
   [-68251/113513400]<br />
</small></code> </p>
<p>β − χ:<br />
<code><small>    [1, -2/3, -1/3, 38/45, -1/3, -3118/4725, 4769/4725, -25666/99225]<br />
   [5/6, -14/15, -7/9, 50/21, -247/270, -14404/4725, 193931/42525]<br />
   [16/15, -34/21, -5/3, 17564/2835, -36521/14175, -1709614/155925]<br />
   [2069/1260, -28/9, -49877/14175, 2454416/155925, -637699/85050]<br />
   [883/315, -28244/4455, -20989/2835, 48124558/1216215]<br />
   [797222/155925, -2471888/184275, -16969807/1091475]<br />
   [2199332/225225, -1238578/42525]<br />
   [87600385/4540536]<br />
</small></code> </p>
<p>χ − θ:<br />
<code><small>    [0, 2/3, 2/3, -2/9, -14/45, 1042/4725, 18/175, -1738/11025]<br />
   [-1/3, 4/15, 43/45, -4/45, -712/945, 332/945, 23159/42525]<br />
   [-2/5, 2/105, 124/105, 274/2835, -1352/945, 13102/31185]<br />
   [-55/126, -16/105, 21068/14175, 1528/4725, -2414843/935550]<br />
   [-22/45, -9202/31185, 20704/10395, 60334/93555]<br />
   [-90263/155925, -299444/675675, 40458083/14189175]<br />
   [-8962/12285, -3818498/6081075]<br />
   [-4259027/4365900]<br />
</small></code> </p>
<p>θ − χ:<br />
<code><small>    [0, -2/3, -2/3, 4/9, 2/9, -3658/4725, 76/225, 64424/99225]<br />
   [1/3, -4/15, -23/45, 68/45, 61/135, -2728/945, 2146/1215]<br />
   [2/5, -24/35, -46/35, 9446/2835, 428/945, -95948/10395]<br />
   [83/126, -80/63, -34712/14175, 4472/525, 29741/85050]<br />
   [52/45, -2362/891, -17432/3465, 280108/13365]<br />
   [335882/155925, -548752/96525, -48965632/4729725]<br />
   [51368/12285, -197456/15795]<br />
   [1461335/174636]<br />
</small></code> </p>
<p>χ − μ:<br />
<code><small>    [-1/2, 2/3, -37/96, 1/360, 81/512, -96199/604800, 5406467/38707200, -7944359/67737600]<br />
   [-1/48, -1/15, 437/1440, -46/105, 1118711/3870720, -51841/1209600, -24749483/348364800]<br />
   [-17/480, 37/840, 209/4480, -5569/90720, -9261899/58060800, 6457463/17740800]<br />
   [-4397/161280, 11/504, 830251/7257600, -466511/2494800, -324154477/7664025600]<br />
   [-4583/161280, 108847/3991680, 8005831/63866880, -22894433/124540416]<br />
   [-20648693/638668800, 16363163/518918400, 2204645983/12915302400]<br />
   [-219941297/5535129600, 497323811/12454041600]<br />
   [-191773887257/3719607091200]<br />
</small></code> </p>
<p>μ − χ:<br />
<code><small>    [1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200]<br />
   [13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400]<br />
   [61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600]<br />
   [49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600]<br />
   [34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080]<br />
   [212378941/319334400, -30705481/10378368, 175214326799/58118860800]<br />
   [1522256789/1383782400, -16759934899/3113510400]<br />
   [1424729850961/743921418240]<br />
</small></code> </p>
<p>ξ − φ:<br />
<code><small>    [-4/3, -4/45, 88/315, 538/4725, 20824/467775, -44732/2837835, -86728/16372125, -88002076/13956067125]<br />
   [34/45, 8/105, -2482/14175, -37192/467775, -12467764/212837625, -895712/147349125, -2641983469/488462349375]<br />
   [-1532/2835, -898/14175, 54968/467775, 100320856/1915538625, 240616/4209975, 8457703444/488462349375]<br />
   [6007/14175, 24496/467775, -5884124/70945875, -4832848/147349125, -4910552477/97692469875]<br />
   [-23356/66825, -839792/19348875, 816824/13395375, 9393713176/488462349375]<br />
   [570284222/1915538625, 1980656/54729675, -4532926649/97692469875]<br />
   [-496894276/1915538625, -14848113968/488462349375]<br />
   [224557742191/976924698750]<br />
</small></code> </p>
<p>φ − ξ:<br />
<code><small>    [4/3, 4/45, -16/35, -2582/14175, 60136/467775, 28112932/212837625, 22947844/1915538625, -1683291094/37574026875]<br />
   [46/45, 152/945, -11966/14175, -21016/51975, 251310128/638512875, 1228352/3007125, -14351220203/488462349375]<br />
   [3044/2835, 3802/14175, -94388/66825, -8797648/10945935, 138128272/147349125, 505559334506/488462349375]<br />
   [6059/4725, 41072/93555, -1472637812/638512875, -45079184/29469825, 973080708361/488462349375]<br />
   [768272/467775, 455935736/638512875, -550000184/147349125, -1385645336626/488462349375]<br />
   [4210684958/1915538625, 443810768/383107725, -2939205114427/488462349375]<br />
   [387227992/127702575, 101885255158/54273594375]<br />
   [1392441148867/325641566250]<br />
</small></code> </p>
<p>ξ − β:<br />
<code><small>    [-1/3, -4/45, 32/315, 34/675, 2476/467775, -70496/8513505, -18484/4343625, 29232878/97692469875]<br />
   [-7/90, -4/315, 74/2025, 3992/467775, 53836/212837625, -4160804/1915538625, -324943819/488462349375]<br />
   [-83/2835, 2/14175, 7052/467775, -661844/1915538625, 237052/383107725, -168643106/488462349375]<br />
   [-797/56700, 934/467775, 1425778/212837625, -2915326/1915538625, 113042383/97692469875]<br />
   [-3673/467775, 390088/212837625, 6064888/1915538625, -558526274/488462349375]<br />
   [-18623681/3831077250, 41288/29469825, 155665021/97692469875]<br />
   [-6205669/1915538625, 504234982/488462349375]<br />
   [-8913001661/3907698795000]<br />
</small></code> </p>
<p>β − ξ:<br />
<code><small>    [1/3, 4/45, -46/315, -1082/14175, 11824/467775, 7947332/212837625, 9708931/1915538625, -5946082372/488462349375]<br />
   [17/90, 68/945, -338/2025, -16672/155925, 39946703/638512875, 164328266/1915538625, 190673521/69780335625]<br />
   [461/2835, 1102/14175, -101069/467775, -255454/1563705, 236067184/1915538625, 86402898356/488462349375]<br />
   [3161/18900, 1786/18711, -189032762/638512875, -98401826/383107725, 110123070361/488462349375]<br />
   [88868/467775, 80274086/638512875, -802887278/1915538625, -200020620676/488462349375]<br />
   [880980241/3831077250, 66263486/383107725, -296107325077/488462349375]<br />
   [37151038/127702575, 4433064236/18091198125]<br />
   [495248998393/1302566265000]<br />
</small></code> </p>
<p>ξ − θ:<br />
<code><small>    [2/3, -4/45, 62/105, 778/4725, -193082/467775, -4286228/42567525, 53702182/212837625, 182466964/8881133625]<br />
   [4/45, -32/315, 12338/14175, 92696/467775, -61623938/70945875, -32500616/273648375, 367082779691/488462349375]<br />
   [-524/2835, -1618/14175, 612536/467775, 427003576/1915538625, -663111728/383107725, -42668482796/488462349375]<br />
   [-5933/14175, -8324/66825, 427770788/212837625, 421877252/1915538625, -327791986997/97692469875]<br />
   [-320044/467775, -9153184/70945875, 6024982024/1915538625, 74612072536/488462349375]<br />
   [-1978771378/1915538625, -46140784/383107725, 489898512247/97692469875]<br />
   [-2926201612/1915538625, -42056042768/488462349375]<br />
   [-2209250801969/976924698750]<br />
</small></code> </p>
<p>θ − ξ:<br />
<code><small>    [-2/3, 4/45, -158/315, -2102/14175, 109042/467775, 216932/2627625, -189115382/1915538625, -230886326/6343666875]<br />
   [16/45, -16/945, 934/14175, -7256/155925, 117952358/638512875, 288456008/1915538625, -11696145869/69780335625]<br />
   [-232/2835, 922/14175, -25286/66825, -7391576/54729675, 478700902/1915538625, 91546732346/488462349375]<br />
   [719/4725, 268/18711, -67048172/638512875, -67330724/383107725, 218929662961/488462349375]<br />
   [14354/467775, 46774256/638512875, -117954842/273648375, -129039188386/488462349375]<br />
   [253129538/1915538625, 2114368/34827975, -178084928947/488462349375]<br />
   [13805944/127702575, 6489189398/54273594375]<br />
   [59983985827/325641566250]<br />
</small></code> </p>
<p>ξ − μ:<br />
<code><small>    [1/6, -4/45, -817/10080, 1297/18900, 7764059/239500800, -9292991/302702400, -25359310709/1743565824000, 39534358147/2858202547200]<br />
   [49/720, -2/35, -29609/453600, 35474/467775, 36019108271/871782912000, -14814966289/245188944000, -13216941177599/571640509440000]<br />
   [4463/90720, -2917/56700, -4306823/59875200, 3026004511/30648618000, 99871724539/1569209241600, -27782109847927/250092722880000]<br />
   [331799/7257600, -102293/1871100, -368661577/4036032000, 2123926699/15324309000, 168979300892599/1600593426432000]<br />
   [11744233/239500800, -875457073/13621608000, -493031379277/3923023104000, 1959350112697/9618950880000]<br />
   [453002260127/7846046208000, -793693009/9807557760, -145659994071373/800296713216000]<br />
   [103558761539/1426553856000, -53583096419057/500185445760000]<br />
   [12272105438887727/128047474114560000]<br />
</small></code> </p>
<p>μ − ξ:<br />
<code><small>    [-1/6, 4/45, 121/1680, -1609/28350, -384229/14968800, 12674323/851350500, 7183403063/560431872000, -375027460897/125046361440000]<br />
   [-29/720, 26/945, 16463/453600, -431/17325, -31621753811/1307674368000, 1117820213/122594472000, 30410873385097/2000741783040000]<br />
   [-1003/45360, 449/28350, 3746047/119750400, -32844781/1751349600, -116359346641/3923023104000, 151567502183/17863765920000]<br />
   [-40457/2419200, 629/53460, 10650637121/326918592000, -13060303/766215450, -317251099510901/8002967132160000]<br />
   [-1800439/119750400, 205072597/20432412000, 146875240637/3923023104000, -2105440822861/125046361440000]<br />
   [-59109051671/3923023104000, 228253559/24518894400, 91496147778023/2000741783040000]<br />
   [-4255034947/261534873600, 126430355893/13894040160000]<br />
   [-791820407649841/42682491371520000]<br />
</small></code> </p>
<p>ξ − χ:<br />
<code><small>    [2/3, -34/45, 46/315, 2458/4725, -55222/93555, 2706758/42567525, 16676974/30405375, -64724382148/97692469875]<br />
   [19/45, -256/315, 3413/14175, 516944/467775, -340492279/212837625, 158999572/1915538625, 85904355287/37574026875]<br />
   [248/567, -15958/14175, 206834/467775, 4430783356/1915538625, -7597644214/1915538625, 2986003168/37574026875]<br />
   [16049/28350, -832976/467775, 62016436/70945875, 851209552/174139875, -375566203/39037950]<br />
   [15602/18711, -651151712/212837625, 3475643362/1915538625, 5106181018156/488462349375]<br />
   [2561772812/1915538625, -10656173804/1915538625, 34581190223/8881133625]<br />
   [873037408/383107725, -5150169424688/488462349375]<br />
   [7939103697617/1953849397500]<br />
</small></code> </p>
<p>χ − ξ:<br />
<code><small>    [-2/3, 34/45, -88/315, -2312/14175, 27128/93555, -55271278/212837625, 308365186/1915538625, -17451293242/488462349375]<br />
   [1/45, -184/945, 6079/14175, -65864/155925, 106691108/638512875, 149984636/1915538625, -101520127208/488462349375]<br />
   [-106/2835, 772/14175, -14246/467775, 5921152/54729675, -99534832/383107725, 10010741462/37574026875]<br />
   [-167/9450, -5312/467775, 75594328/638512875, -35573728/273648375, 1615002539/75148053750]<br />
   [-248/13365, 2837636/638512875, 130601488/1915538625, -3358119706/488462349375]<br />
   [-34761247/1915538625, -3196/3553875, 46771947158/488462349375]<br />
   [-2530364/127702575, -18696014/18091198125]<br />
   [-14744861191/651283132500]<br />
</small></code></p>
<h1><a class="anchor" id="auxlaterror"></a>
Truncation errors</h1>
<p>There are two sources of error when using these series. The truncation error arises from retaing terms up to a certain order in <em>n</em>; it is the absolute difference between the value of the truncated series compared with the exact latitude (evaluated with exact arithmetic). In addition, using standard double-precision arithmetic entails accumulating round-off errors so that at the end of a complex calculation a few of the trailing bits of the result are wrong.</p>
<p>Here's a table of the truncation errors. The errors are given in "units
in the last place (ulp)" where 1 ulp = 2<sup>−53</sup> radian = 6.4 × 10<sup>−15</sup> degree = 2.3 × 10<sup>−11</sup> arcsecond which is a measure of the round-off error for double precision. Here is some rough guidance on how to interpret these errors:</p><ul>
<li>if the truncation error is less than 1 ulp, then round-off errors dominate;</li>
<li>if the truncation error is greater than 8 ulp, then truncation errors dominate;</li>
<li>otherwise, round-off and truncation errors are comparable.</li>
</ul>
<p>The truncation errors are given accurate to 2 significant figures.</p>
<center> <a class="anchor" id=""></a>
<table class="doxtable">
<caption>Auxiliary latitude truncation errors (ulp)</caption>
<tr>
<th rowspan="2">expression </th><th colspan="2">[<em>f</em> = 1/150, order = 6] </th><th colspan="2">[<em>f</em> = 1/297, order = 5] </th></tr>
<tr>
<th><em>n</em> series </th><th><em>e</em><sup>2</sup> series </th><th><em>n</em> series </th><th><em>e</em><sup>2</sup> series </th></tr>
<tr>
<td>β − φ </td><td><code><center> 0.0060  </center></code></td><td><code><center> 28    </center></code></td><td><code><center>  0.035  </center></code></td><td><code><center>  41    </center></code> </td></tr>
<tr>
<td>φ − β </td><td><code><center> 0.0060  </center></code></td><td><code><center> 28    </center></code></td><td><code><center>  0.035  </center></code></td><td><code><center>  41    </center></code> </td></tr>
<tr>
<td>θ − φ </td><td><code><center> 2.9     </center></code></td><td><code><center> 82    </center></code></td><td><code><center>  6.0    </center></code></td><td><code><center> 120    </center></code> </td></tr>
<tr>
<td>φ − θ </td><td><code><center> 2.9     </center></code></td><td><code><center> 82    </center></code></td><td><code><center>  6.0    </center></code></td><td><code><center> 120    </center></code> </td></tr>
<tr>
<td>θ − β </td><td><code><center> 0.0060  </center></code></td><td><code><center> 28    </center></code></td><td><code><center>  0.035  </center></code></td><td><code><center>  41    </center></code> </td></tr>
<tr>
<td>β − θ </td><td><code><center> 0.0060  </center></code></td><td><code><center> 28    </center></code></td><td><code><center>  0.035  </center></code></td><td><code><center>  41    </center></code> </td></tr>
<tr>
<td>μ − φ </td><td><code><center> 0.037   </center></code></td><td><code><center> 41    </center></code></td><td><code><center>  0.18   </center></code></td><td><code><center>  60    </center></code> </td></tr>
<tr>
<td>φ − μ </td><td><code><center> 0.98    </center></code></td><td><code><center> 59    </center></code></td><td><code><center>  2.3    </center></code></td><td><code><center>  84    </center></code> </td></tr>
<tr>
<td>μ − β </td><td><code><center> 0.00069 </center></code></td><td><code><center>  5.8  </center></code></td><td><code><center>  0.0024 </center></code></td><td><code><center>   9.6  </center></code> </td></tr>
<tr>
<td>β − μ </td><td><code><center> 0.13    </center></code></td><td><code><center> 12    </center></code></td><td><code><center>  0.35   </center></code></td><td><code><center>  19    </center></code> </td></tr>
<tr>
<td>μ − θ </td><td><code><center> 0.24    </center></code></td><td><code><center> 30    </center></code></td><td><code><center>  0.67   </center></code></td><td><code><center>  40    </center></code> </td></tr>
<tr>
<td>θ − μ </td><td><code><center> 0.099   </center></code></td><td><code><center> 23    </center></code></td><td><code><center>  0.23   </center></code></td><td><code><center>  33    </center></code> </td></tr>
<tr>
<td>χ − φ </td><td><code><center> 0.78    </center></code></td><td><code><center> 43    </center></code></td><td><code><center>  2.1    </center></code></td><td><code><center>  64    </center></code> </td></tr>
<tr>
<td>φ − χ </td><td><code><center> 9.0     </center></code></td><td><code><center> 71    </center></code></td><td><code><center> 17      </center></code></td><td><code><center> 100    </center></code> </td></tr>
<tr>
<td>χ − β </td><td><code><center> 0.018   </center></code></td><td><code><center>  3.7  </center></code></td><td><code><center>  0.11   </center></code></td><td><code><center>   6.4  </center></code> </td></tr>
<tr>
<td>β − χ </td><td><code><center> 1.7     </center></code></td><td><code><center> 16    </center></code></td><td><code><center>  3.4    </center></code></td><td><code><center>  24    </center></code> </td></tr>
<tr>
<td>χ − θ </td><td><code><center> 0.18    </center></code></td><td><code><center> 31    </center></code></td><td><code><center>  0.56   </center></code></td><td><code><center>  43    </center></code> </td></tr>
<tr>
<td>θ − χ </td><td><code><center> 0.87    </center></code></td><td><code><center> 23    </center></code></td><td><code><center>  1.9    </center></code></td><td><code><center>  32    </center></code> </td></tr>
<tr>
<td>χ − μ </td><td><code><center> 0.022   </center></code></td><td><code><center>  0.56 </center></code></td><td><code><center>  0.11   </center></code></td><td><code><center>   0.91 </center></code> </td></tr>
<tr>
<td>μ − χ </td><td><code><center> 0.31    </center></code></td><td><code><center>  1.2  </center></code></td><td><code><center>  0.86   </center></code></td><td><code><center>   2.0  </center></code> </td></tr>
<tr>
<td>ξ − φ </td><td><code><center> 0.015   </center></code></td><td><code><center> 39    </center></code></td><td><code><center>  0.086  </center></code></td><td><code><center>  57    </center></code> </td></tr>
<tr>
<td>φ − ξ </td><td><code><center> 0.34    </center></code></td><td><code><center> 53    </center></code></td><td><code><center>  1.1    </center></code></td><td><code><center>  75    </center></code> </td></tr>
<tr>
<td>ξ − β </td><td><code><center> 0.00042 </center></code></td><td><code><center>  6.3  </center></code></td><td><code><center>  0.0039 </center></code></td><td><code><center>  10    </center></code> </td></tr>
<tr>
<td>β − ξ </td><td><code><center> 0.040   </center></code></td><td><code><center> 10    </center></code></td><td><code><center>  0.15   </center></code></td><td><code><center>  15    </center></code> </td></tr>
<tr>
<td>ξ − θ </td><td><code><center> 0.28    </center></code></td><td><code><center> 28    </center></code></td><td><code><center>  0.75   </center></code></td><td><code><center>  38    </center></code> </td></tr>
<tr>
<td>θ − ξ </td><td><code><center> 0.040   </center></code></td><td><code><center> 23    </center></code></td><td><code><center>  0.11   </center></code></td><td><code><center>  33    </center></code> </td></tr>
<tr>
<td>ξ − μ </td><td><code><center> 0.015   </center></code></td><td><code><center>  0.79 </center></code></td><td><code><center>  0.058  </center></code></td><td><code><center>   1.5  </center></code> </td></tr>
<tr>
<td>μ − ξ </td><td><code><center> 0.0043  </center></code></td><td><code><center>  0.54 </center></code></td><td><code><center>  0.018  </center></code></td><td><code><center>   1.1  </center></code> </td></tr>
<tr>
<td>ξ − χ </td><td><code><center> 0.60    </center></code></td><td><code><center>  1.9  </center></code></td><td><code><center>  1.5    </center></code></td><td><code><center>   3.6  </center></code> </td></tr>
<tr>
<td>χ − ξ </td><td><code><center> 0.023   </center></code></td><td><code><center>  0.53 </center></code></td><td><code><center>  0.079  </center></code></td><td><code><center>   0.92 </center></code> </td></tr>
</table>
</center><p>The 2nd and 3rd columns show the results for the SRMmax ellipsoid, <em>f</em> = 1/150, retaining 6th order terms in the series expansion. The 4th and 5th columns show the results for the International ellipsoid, <em>f</em> = 1/297, retaining 5th order terms in the series expansion. The 2nd and 4th columns give the errors for the series expansions in terms of <em>n</em> given in this section (appropriately truncated). The 3rd and 5th columns give the errors when the series are reexpanded in terms of <em>e</em><sup>2</sup> = 4<em>n/</em>(1 + <em>n</em>)<sup>2</sup> and truncated retaining the <em>e</em><sup>12</sup> and <em>e</em><sup>10</sup> terms respectively.</p>
<p>Some observations:</p><ul>
<li>For production use, the 6th order series in <em>n</em> are recommended. For <em>f</em> = 1/150, the resulting errors are close to the round-off limit. The errors in the 6th order series scale as <em>f</em><sup>7</sup>; so the errors with <em>f</em> = 1/297 are about 120 times smaller.</li>
<li>It's inadvisable to use the 5th order series in <em>n</em>; this order is barely acceptable for <em>f</em> = 1/297 and the errors grow as <em>f</em><sup>6</sup> as <em>f</em> is increased.</li>
<li>In all cases, the expansions in terms of <em>e</em><sup>2</sup> are considerably less accurate than the corresponding series in <em>n</em>.</li>
<li>For every series converting between φ and any of θ, μ, χ, or ξ, the series where β is substituted for φ is more accurate. Considering that the transformation between φ and β is so simple, tanβ = (1 - <em>f</em>) tanφ, it sometimes makes sense to use β internally as the basic measure of latitude. (This is the case with geodesic calculations.)</li>
</ul>
<center> Back to <a class="el" href="geocentric.html">Geocentric coordinates</a>. Forward to <a class="el" href="highprec.html">Support for high precision arithmetic</a>. Up to <a class="el" href="index.html#contents">Contents</a>. </center> </div></div><!-- contents -->
<!-- start footer part -->
<hr class="footer"/><address class="footer"><small>
Generated by  <a href="http://www.doxygen.org/index.html">
<img class="footer" src="doxygen.png" alt="doxygen"/>
</a> 1.8.13
</small></address>
</body>
</html>
|