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href="../index.html">Coordinate operations</a> »</li> + <li><a href="index.html">Transformations</a> »</li> + <li>Horner polynomial evaluation</li> + <li class="wy-breadcrumbs-aside"> + <a href="https://github.com/OSGeo/PROJ/edit/8.2/docs/source/operations/transformations/horner.rst" class="fa fa-github"> Edit on GitHub</a> + </li> + </ul><div class="rst-breadcrumbs-buttons" role="navigation" aria-label="Sequential page navigation"> + <a href="helmert.html" class="btn btn-neutral float-left" title="Helmert transform" accesskey="p"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="molodensky.html" class="btn btn-neutral float-right" title="Molodensky transform" accesskey="n">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + <hr/> +</div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="horner-polynomial-evaluation"> +<span id="horner"></span><h1>Horner polynomial evaluation<a class="headerlink" href="#horner-polynomial-evaluation" title="Permalink to this headline">¶</a></h1> +<div class="versionadded"> +<p><span class="versionmodified added">New in version 5.0.0.</span></p> +</div> +<table class="docutils align-default"> +<colgroup> +<col style="width: 20%" /> +<col style="width: 80%" /> +</colgroup> +<tbody> +<tr class="row-odd"><td><p><strong>Alias</strong></p></td> +<td><p>horner</p></td> +</tr> +<tr class="row-even"><td><p><strong>Domain</strong></p></td> +<td><p>2D and 3D</p></td> +</tr> +<tr class="row-odd"><td><p><strong>Input type</strong></p></td> +<td><p>Geodetic and projected coordinates</p></td> +</tr> +<tr class="row-even"><td><p><strong>Output type</strong></p></td> +<td><p>Geodetic and projected coordinates</p></td> +</tr> +</tbody> +</table> +<p>The Horner polynomial evaluation scheme is used for transformations between reference +frames where one or both are inhomogeneous or internally distorted. This will typically +be reference frames created before the introduction of space geodetic techniques such +as GPS.</p> +<p>Horner polynomials, or Multiple Regression Equations as they are also known as, have +their strength in being able to create complicated mappings between coordinate +reference frames while still being lightweight in both computational cost and disk +space used.</p> +<p>PROJ implements two versions of the Horner evaluation scheme: Real and complex +polynomial evaluation. Below both are briefly described. For more details consult +<span id="id1">[<a class="reference internal" href="../../zreferences.html#id31" title="Ruffhead, A. C. Introduction to multiple regression equations in datum transformations and their reversibility. Survey Review, 50(358):82–90, 2016. doi:10.1080/00396265.2016.1244143.">Ruffhead2016</a>]</span> and <span id="id2">[<a class="reference internal" href="../../zreferences.html#id17" title="IOGP. Geomatics guidance note 7, part 2: coordinate conversions & transformations including formulas. IOGP Publication 373-7-2, International Association For Oil And Gas Producers, 2018. URL: https://www.iogp.org/bookstore/product/coordinate-conversions-and-transformation-including-formulas/.">IOGP2018</a>]</span>.</p> +<p>The polynomial evaluation in real number space is defined by the following equations:</p> +<div class="math notranslate nohighlight" id="equation-real-poly"> +<span class="eqno">(1)<a class="headerlink" href="#equation-real-poly" title="Permalink to this equation">¶</a></span>\[ \begin{align}\begin{aligned}\Delta X = \sum_{i,j} u_{i,j} U^i V^j\\\Delta Y = \sum_{i,j} v_{i,j} U^i V^j\end{aligned}\end{align} \]</div> +<p>where</p> +<div class="math notranslate nohighlight" id="equation-uv"> +<span class="eqno">(2)<a class="headerlink" href="#equation-uv" title="Permalink to this equation">¶</a></span>\[ \begin{align}\begin{aligned}U = X_{in} - X_{origin}\\V = Y_{in} - Y_{origin}\end{aligned}\end{align} \]</div> +<p>and <span class="math notranslate nohighlight">\(u_{i,j}\)</span> and <span class="math notranslate nohighlight">\(v_{i,j}\)</span> are coefficients that make up +the polynomial.</p> +<p>The final coordinates are determined as</p> +<div class="math notranslate nohighlight" id="equation-xy-out"> +<span class="eqno">(3)<a class="headerlink" href="#equation-xy-out" title="Permalink to this equation">¶</a></span>\[ \begin{align}\begin{aligned}X_{out} = X_{in} + \Delta X\\Y_{out} = Y_{in} + \Delta Y\end{aligned}\end{align} \]</div> +<p>The inverse transform is the same as the above but requires a different set of +coefficients.</p> +<p>Evaluation of the complex polynomials are defined by the following equations:</p> +<div class="math notranslate nohighlight" id="equation-complex-poly"> +<span class="eqno">(4)<a class="headerlink" href="#equation-complex-poly" title="Permalink to this equation">¶</a></span>\[\Delta X + i \Delta Y = \sum_{j=1}^n (c_{2j-1} + i c_{2j}) (U + i V)^j\]</div> +<p>Where <span class="math notranslate nohighlight">\(n\)</span> is the degree of the polynomial. <span class="math notranslate nohighlight">\(U\)</span> and <span class="math notranslate nohighlight">\(V\)</span> are +defined as in <a class="reference internal" href="#equation-uv">(2)</a> and the resulting coordinates are again determined +by <a class="reference internal" href="#equation-xy-out">(3)</a>.</p> +<section id="examples"> +<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2> +<p>Mapping between Danish TC32 and ETRS89/UTM zone 32 using polynomials in real +number space:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>+proj=horner ++ellps=intl ++range=500000 ++fwd_origin=877605.269066,6125810.306769 ++inv_origin=877605.760036,6125811.281773 ++deg=4 ++fwd_v=6.1258112678e+06,9.9999971567e-01,1.5372750011e-10,5.9300860915e-15,2.2609497633e-19,4.3188227445e-05,2.8225130416e-10,7.8740007114e-16,-1.7453997279e-19,1.6877465415e-10,-1.1234649773e-14,-1.7042333358e-18,-7.9303467953e-15,-5.2906832535e-19,3.9984284847e-19 ++fwd_u=8.7760574982e+05,9.9999752475e-01,2.8817299305e-10,5.5641310680e-15,-1.5544700949e-18,-4.1357045890e-05,4.2106213519e-11,2.8525551629e-14,-1.9107771273e-18,3.3615590093e-10,2.4380247154e-14,-2.0241230315e-18,1.2429019719e-15,5.3886155968e-19,-1.0167505000e-18 ++inv_v=6.1258103208e+06,1.0000002826e+00,-1.5372762184e-10,-5.9304261011e-15,-2.2612705361e-19,-4.3188331419e-05,-2.8225549995e-10,-7.8529116371e-16,1.7476576773e-19,-1.6875687989e-10,1.1236475299e-14,1.7042518057e-18,7.9300735257e-15,5.2881862699e-19,-3.9990736798e-19 ++inv_u=8.7760527928e+05,1.0000024735e+00,-2.8817540032e-10,-5.5627059451e-15,1.5543637570e-18,4.1357152105e-05,-4.2114813612e-11,-2.8523713454e-14,1.9109017837e-18,-3.3616407783e-10,-2.4382678126e-14,2.0245020199e-18,-1.2441377565e-15,-5.3885232238e-19,1.0167203661e-18 +</pre></div> +</div> +<p>Mapping between Danish System Storebælt and ETRS89/UTM zone 32 using complex +polynomials:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>+proj=horner ++ellps=intl ++range=500000 ++fwd_origin=4.94690026817276e+05,6.13342113183056e+06 ++inv_origin=6.19480258923588e+05,6.13258568148837e+06 ++deg=3 ++fwd_c=6.13258562111350e+06,6.19480105709997e+05,9.99378966275206e-01,-2.82153291753490e-02,-2.27089979140026e-10,-1.77019590701470e-09,1.08522286274070e-14,2.11430298751604e-15 ++inv_c=6.13342118787027e+06,4.94690181709311e+05,9.99824464710368e-01,2.82279070814774e-02,7.66123542220864e-11,1.78425334628927e-09,-1.05584823306400e-14,-3.32554258683744e-15 +</pre></div> +</div> +</section> +<section id="parameters"> +<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2> +<p>Setting up Horner polynomials requires many coefficients being explicitly +written, even for polynomials of low degree. For this reason it is recommended +to store the polynomial definitions in an <a class="reference internal" href="../../resource_files.html#init-files"><span class="std std-ref">init file</span></a> for +easier writing and reuse.</p> +<section id="required"> +<h3>Required<a class="headerlink" href="#required" title="Permalink to this headline">¶</a></h3> +<p>Below is a list of required parameters that can be set for the Horner polynomial +transformation. As stated above, the transformation takes to forms, either using +real or complex polynomials. These are divided into separate sections below. +Parameters from the two sections are mutually exclusive, that is parameters +describing real and complex polynomials can’t be mixed.</p> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-ellps"> +<span class="sig-name descname"><span class="pre">+ellps</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-ellps" title="Permalink to this definition">¶</a></dt> +<dd><p>The name of a built-in ellipsoid definition.</p> +<p>See <a class="reference internal" href="../../usage/ellipsoids.html#ellipsoids"><span class="std std-ref">Ellipsoids</span></a> for more information, or execute +<a class="reference internal" href="../../apps/proj.html#cmdoption-proj-le"><code class="xref std std-option docutils literal notranslate"><span class="pre">proj</span> <span class="pre">-le</span></code></a> for a list of built-in ellipsoid names.</p> +<p><em>Defaults to “GRS80”.</em></p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-deg"> +<span class="sig-name descname"><span class="pre">+deg</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-deg" title="Permalink to this definition">¶</a></dt> +<dd><p>Degree of polynomial</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-fwd_origin"> +<span id="cmdoption-arg-fwd-origin"></span><span class="sig-name descname"><span class="pre">+fwd_origin</span></span><span class="sig-prename descclassname"><span class="pre">=<northing,easting></span></span><a class="headerlink" href="#cmdoption-arg-fwd_origin" title="Permalink to this definition">¶</a></dt> +<dd><p>Coordinate of origin for the forward mapping</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-inv_origin"> +<span id="cmdoption-arg-inv-origin"></span><span class="sig-name descname"><span class="pre">+inv_origin</span></span><span class="sig-prename descclassname"><span class="pre">=<northing,easting></span></span><a class="headerlink" href="#cmdoption-arg-inv_origin" title="Permalink to this definition">¶</a></dt> +<dd><p>Coordinate of origin for the inverse mapping</p> +</dd></dl> + +<section id="real-polynomials"> +<h4>Real polynomials<a class="headerlink" href="#real-polynomials" title="Permalink to this headline">¶</a></h4> +<p>The following parameters has to be set if the transformation consists of +polynomials in real space. Each parameter takes a comma-separated list of +coefficients. The number of coefficients is governed by the degree, <span class="math notranslate nohighlight">\(d\)</span>, +of the polynomial:</p> +<div class="math notranslate nohighlight"> +\[N = \frac{(d + 1)(d + 2)}{2}\]</div> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-fwd_u"> +<span id="cmdoption-arg-fwd-u"></span><span class="sig-name descname"><span class="pre">+fwd_u</span></span><span class="sig-prename descclassname"><span class="pre">=<u_11,u_12,...,u_ij,..,u_mn></span></span><a class="headerlink" href="#cmdoption-arg-fwd_u" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the forward transformation i.e. latitude to northing +as described in <a class="reference internal" href="#equation-real-poly">(1)</a>.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-fwd_v"> +<span id="cmdoption-arg-fwd-v"></span><span class="sig-name descname"><span class="pre">+fwd_v</span></span><span class="sig-prename descclassname"><span class="pre">=<v_11,v_12,...,v_ij,..,v_mn></span></span><a class="headerlink" href="#cmdoption-arg-fwd_v" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the forward transformation i.e. longitude to easting +as described in <a class="reference internal" href="#equation-real-poly">(1)</a>.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-inv_u"> +<span id="cmdoption-arg-inv-u"></span><span class="sig-name descname"><span class="pre">+inv_u</span></span><span class="sig-prename descclassname"><span class="pre">=<u_11,u_12,...,u_ij,..,u_mn></span></span><a class="headerlink" href="#cmdoption-arg-inv_u" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the inverse transformation i.e. latitude to northing +as described in <a class="reference internal" href="#equation-real-poly">(1)</a>.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-inv_v"> +<span id="cmdoption-arg-inv-v"></span><span class="sig-name descname"><span class="pre">+inv_v</span></span><span class="sig-prename descclassname"><span class="pre">=<v_11,v_12,...,v_ij,..,v_mn></span></span><a class="headerlink" href="#cmdoption-arg-inv_v" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the inverse transformation i.e. longitude to easting +as described in <a class="reference internal" href="#equation-real-poly">(1)</a>.</p> +</dd></dl> + +</section> +<section id="complex-polynomials"> +<h4>Complex polynomials<a class="headerlink" href="#complex-polynomials" title="Permalink to this headline">¶</a></h4> +<p>The following parameters has to be set if the transformation consists of +polynomials in complex space. Each parameter takes a comma-separated list of +coefficients. The number of coefficients is governed by the degree, <span class="math notranslate nohighlight">\(d\)</span>, +of the polynomial:</p> +<div class="math notranslate nohighlight"> +\[N = 2d + 2\]</div> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-fwd_c"> +<span id="cmdoption-arg-fwd-c"></span><span class="sig-name descname"><span class="pre">+fwd_c</span></span><span class="sig-prename descclassname"><span class="pre">=<c_1,c_2,...,c_N></span></span><a class="headerlink" href="#cmdoption-arg-fwd_c" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the complex forward transformation +as described in <a class="reference internal" href="#equation-complex-poly">(4)</a>.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-inv_c"> +<span id="cmdoption-arg-inv-c"></span><span class="sig-name descname"><span class="pre">+inv_c</span></span><span class="sig-prename descclassname"><span class="pre">=<c_1,c_2,...,c_N></span></span><a class="headerlink" href="#cmdoption-arg-inv_c" title="Permalink to this definition">¶</a></dt> +<dd><p>Coefficients for the complex inverse transformation +as described in <a class="reference internal" href="#equation-complex-poly">(4)</a>.</p> +</dd></dl> + +</section> +</section> +<section id="optional"> +<h3>Optional<a class="headerlink" href="#optional" title="Permalink to this headline">¶</a></h3> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-range"> +<span class="sig-name descname"><span class="pre">+range</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-range" title="Permalink to this definition">¶</a></dt> +<dd><p>Radius of the region of validity.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-uneg"> +<span class="sig-name descname"><span class="pre">+uneg</span></span><span class="sig-prename descclassname"></span><a class="headerlink" href="#cmdoption-arg-uneg" title="Permalink to this definition">¶</a></dt> +<dd><p>Express latitude as southing. Only applies for complex polynomials.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-vneg"> +<span class="sig-name descname"><span class="pre">+vneg</span></span><span class="sig-prename descclassname"></span><a class="headerlink" href="#cmdoption-arg-vneg" title="Permalink to this definition">¶</a></dt> +<dd><p>Express longitude as westing. Only applies for complex polynomials.</p> +</dd></dl> + +</section> +</section> +<section id="further-reading"> +<h2>Further reading<a class="headerlink" href="#further-reading" title="Permalink to this headline">¶</a></h2> +<ol class="arabic simple"> +<li><p><a class="reference external" href="https://en.wikipedia.org/wiki/Horner%27s_method">Wikipedia</a></p></li> +</ol> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="helmert.html" class="btn btn-neutral float-left" title="Helmert transform" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="molodensky.html" class="btn btn-neutral float-right" title="Molodensky transform" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + + <hr/> + + <div role="contentinfo"> + <p>© Copyright 1983-2022. + <span class="lastupdated">Last updated on 22 Mar 2022. + </span></p> + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + +</footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + +</body> +</html>
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