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  <section id="molodensky-badekas-transform">
<span id="molobadekas"></span><h1>Molodensky-Badekas transform<a class="headerlink" href="#molodensky-badekas-transform" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
<p><span class="versionmodified added">New in version 6.0.0.</span></p>
</div>
<p>The Molodensky-Badekas transformation changes coordinates from one reference frame to
another by means of a 10-parameter shift.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>It should not be confused with the <a class="reference internal" href="molodensky.html#molodensky"><span class="std std-ref">Molodensky transform</span></a> transform which
operates directly in the geodetic coordinates. Molodensky-Badekas can rather
be seen as a variation of <a class="reference internal" href="helmert.html#helmert"><span class="std std-ref">Helmert transform</span></a></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>molobadekas</p></td>
</tr>
<tr class="row-even"><td><p><strong>Domain</strong></p></td>
<td><p>3D</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Input type</strong></p></td>
<td><p>Cartesian coordinates</p></td>
</tr>
<tr class="row-even"><td><p><strong>Output type</strong></p></td>
<td><p>Cartesian coordinates</p></td>
</tr>
</tbody>
</table>
<p>The Molodensky-Badekas transformation is a variation of the <a class="reference internal" href="helmert.html#helmert"><span class="std std-ref">Helmert transform</span></a> where
the rotational terms are not directly applied to the ECEF coordinates, but on
cartesian coordinates relative to a reference point (usually close to Earth surface,
and to the area of use of the transformation). When <code class="docutils literal notranslate"><span class="pre">px</span></code> = <code class="docutils literal notranslate"><span class="pre">py</span></code> = <code class="docutils literal notranslate"><span class="pre">pz</span></code> = 0,
this is equivalent to a 7-parameter Helmert transformation.</p>
<section id="example">
<h2>Example<a class="headerlink" href="#example" title="Permalink to this headline">¶</a></h2>
<p>Transforming coordinates from La Canoa to REGVEN:</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=molobadekas convention=coordinate_frame
       x=-270.933 y=115.599 z=-360.226 rx=-5.266 ry=-1.238 rz=2.381
       s=-5.109 px=2464351.59 py=-5783466.61 pz=974809.81
</pre></div>
</div>
</section>
<section id="parameters">
<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>All parameters (except convention) are optional but at least one should be
used, otherwise the operation will return the coordinates unchanged.</p>
</div>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-convention">
<span class="sig-name descname"><span class="pre">+convention</span></span><span class="sig-prename descclassname"><span class="pre">=coordinate_frame/position_vector</span></span><a class="headerlink" href="#cmdoption-arg-convention" title="Permalink to this definition">¶</a></dt>
<dd><p>Indicates the convention to express the rotational terms when a 3D-Helmert /
7-parameter more transform is involved.</p>
<p>The two conventions are equally popular and a frequent source of confusion.
The coordinate frame convention is also described as an clockwise
rotation of the coordinate frame. It corresponds to EPSG method code
1034 (in the geocentric domain) or 9636 (in the geographic domain)
The position vector convention is also described as an anticlockwise
(counter-clockwise) rotation of the coordinate frame.
It corresponds to as EPSG method code 1061 (in the geocentric domain) or
1063 (in the geographic domain).</p>
<p>The result obtained with parameters specified in a given convention
can be obtained in the other convention by negating the rotational parameters
(<code class="docutils literal notranslate"><span class="pre">rx</span></code>, <code class="docutils literal notranslate"><span class="pre">ry</span></code>, <code class="docutils literal notranslate"><span class="pre">rz</span></code>)</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-x">
<span class="sig-name descname"><span class="pre">+x</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-x" title="Permalink to this definition">¶</a></dt>
<dd><p>Translation of the x-axis given in meters.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-y">
<span class="sig-name descname"><span class="pre">+y</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-y" title="Permalink to this definition">¶</a></dt>
<dd><p>Translation of the y-axis given in meters.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-z">
<span class="sig-name descname"><span class="pre">+z</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-z" title="Permalink to this definition">¶</a></dt>
<dd><p>Translation of the z-axis given in meters.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-s">
<span class="sig-name descname"><span class="pre">+s</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-s" title="Permalink to this definition">¶</a></dt>
<dd><p>Scale factor given in ppm.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-rx">
<span class="sig-name descname"><span class="pre">+rx</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-rx" title="Permalink to this definition">¶</a></dt>
<dd><p>X-axis rotation given arc seconds.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-ry">
<span class="sig-name descname"><span class="pre">+ry</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-ry" title="Permalink to this definition">¶</a></dt>
<dd><p>Y-axis rotation given in arc seconds.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-rz">
<span class="sig-name descname"><span class="pre">+rz</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-rz" title="Permalink to this definition">¶</a></dt>
<dd><p>Z-axis rotation given in arc seconds.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-px">
<span class="sig-name descname"><span class="pre">+px</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-px" title="Permalink to this definition">¶</a></dt>
<dd><p>Coordinate along the x-axis of the reference point given in meters.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-py">
<span class="sig-name descname"><span class="pre">+py</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-py" title="Permalink to this definition">¶</a></dt>
<dd><p>Coordinate along the y-axis of the reference point given in meters.</p>
</dd></dl>

<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-pz">
<span class="sig-name descname"><span class="pre">+pz</span></span><span class="sig-prename descclassname"><span class="pre">=&lt;value&gt;</span></span><a class="headerlink" href="#cmdoption-arg-pz" title="Permalink to this definition">¶</a></dt>
<dd><p>Coordinate along the z-axis of the reference point given in meters.</p>
</dd></dl>

</section>
<section id="mathematical-description">
<h2>Mathematical description<a class="headerlink" href="#mathematical-description" title="Permalink to this headline">¶</a></h2>
<p>In the <em>Position Vector</em> convention, we define <span class="math notranslate nohighlight">\(R_x = radians \left( rx \right)\)</span>,
<span class="math notranslate nohighlight">\(R_z = radians \left( ry \right)\)</span> and <span class="math notranslate nohighlight">\(R_z = radians \left( rz \right)\)</span></p>
<p>In the <em>Coordinate Frame</em> convention, <span class="math notranslate nohighlight">\(R_x = - radians \left( rx \right)\)</span>,
<span class="math notranslate nohighlight">\(R_z = - radians \left( ry \right)\)</span> and <span class="math notranslate nohighlight">\(R_z = - radians \left( rz \right)\)</span></p>
<div class="math notranslate nohighlight" id="equation-10param">
<span class="eqno">(1)<a class="headerlink" href="#equation-10param" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{align}
    \begin{bmatrix}
        X \\
        Y \\
        Z \\
    \end{bmatrix}^{output} =
    \begin{bmatrix}
        T_x + P_x\\
        T_y + P_y\\
        T_z + P_z\\
    \end{bmatrix} +
    \left(1 + s \times 10^{-6}\right)
    \begin{bmatrix}
         1  &amp; -R_z  &amp;  R_y \\
         Rz &amp;  1    &amp; -R_x \\
        -Ry &amp;  R_x  &amp;  1   \\
    \end{bmatrix}
    \begin{bmatrix}
        X^{input} - P_x\\
        Y^{input} - P_y\\
        Z^{input} - P_z\\
    \end{bmatrix}
\end{align}\end{split}\]</div>
</section>
</section>


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