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<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>
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