/usr/share/doc/geographiclib/html/auxlat.html is in geographiclib-doc 1.45-2.
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 | <!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 1.8.9.1"/>
<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 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 style="padding-left: 0.5em;">
<div id="projectname">GeographicLib
 <span id="projectnumber">1.45</span>
</div>
</td>
</tr>
</tbody>
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.8.9.1 -->
<div id="navrow1" class="tabs">
<ul class="tablist">
<li><a href="index.html"><span>Main Page</span></a></li>
<li class="current"><a href="pages.html"><span>Related Pages</span></a></li>
<li><a href="namespaces.html"><span>Namespaces</span></a></li>
<li><a href="annotated.html"><span>Classes</span></a></li>
<li><a href="files.html"><span>Files</span></a></li>
</ul>
</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>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>. (In addition there's the isometric latitude, ψ = sinh<sup>−1</sup> tanχ; but this is not an angle-like variable and we don't consider it further here.) 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 Clenshaw summation 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. 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 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, ∞). (Some authors use the eccentricity as the expansion parameter, but the resulting series only converge for <em>b</em>/<em>a</em> ∈ (0, √2).) These expansions were obtained with the the maxima code, <a href="auxlat.mac">auxlat.mac</a>.</p>
<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>
<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>
<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><center></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> 1.8.9.1
</small></address>
</body>
</html>
|