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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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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">=<value></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>\[\begin{split}\begin{align} + \begin{bmatrix} + X \\ + Y \\ + Z \\ + \end{bmatrix}^{output} = + \begin{bmatrix} + T_x + P_x\\ + T_y + P_y\\ + T_z + P_z\\ + \end{bmatrix} + + \left(1 + s \times 10^{-6}\right) + \begin{bmatrix} + 1 & -R_z & R_y \\ + Rz & 1 & -R_x \\ + -Ry & R_x & 1 \\ + \end{bmatrix} + \begin{bmatrix} + X^{input} - P_x\\ + Y^{input} - P_y\\ + Z^{input} - P_z\\ + \end{bmatrix} +\end{align}\end{split}\]</div> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="molodensky.html" class="btn btn-neutral float-left" title="Molodensky transform" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="hgridshift.html" class="btn btn-neutral float-right" title="Horizontal grid shift" 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 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