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<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.Arcball"><code class="docutils literal notranslate"><span class="pre">Arcball</span></code></a></li>
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<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.arcball_map_to_sphere"><code class="docutils literal notranslate"><span class="pre">arcball_map_to_sphere()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.arcball_nearest_axis"><code class="docutils literal notranslate"><span class="pre">arcball_nearest_axis()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.clip_matrix"><code class="docutils literal notranslate"><span class="pre">clip_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.compose_matrix"><code class="docutils literal notranslate"><span class="pre">compose_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.concatenate_matrices"><code class="docutils literal notranslate"><span class="pre">concatenate_matrices()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.decompose_matrix"><code class="docutils literal notranslate"><span class="pre">decompose_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.euler_from_matrix"><code class="docutils literal notranslate"><span class="pre">euler_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.euler_from_quaternion"><code class="docutils literal notranslate"><span class="pre">euler_from_quaternion()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.euler_matrix"><code class="docutils literal notranslate"><span class="pre">euler_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.identity_matrix"><code class="docutils literal notranslate"><span class="pre">identity_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.inverse_matrix"><code class="docutils literal notranslate"><span class="pre">inverse_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.is_same_transform"><code class="docutils literal notranslate"><span class="pre">is_same_transform()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.orthogonalization_matrix"><code class="docutils literal notranslate"><span class="pre">orthogonalization_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.projection_from_matrix"><code class="docutils literal notranslate"><span class="pre">projection_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.projection_matrix"><code class="docutils literal notranslate"><span class="pre">projection_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_about_axis"><code class="docutils literal notranslate"><span class="pre">quaternion_about_axis()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_conjugate"><code class="docutils literal notranslate"><span class="pre">quaternion_conjugate()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_from_euler"><code class="docutils literal notranslate"><span class="pre">quaternion_from_euler()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_from_matrix"><code class="docutils literal notranslate"><span class="pre">quaternion_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_imag"><code class="docutils literal notranslate"><span class="pre">quaternion_imag()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_inverse"><code class="docutils literal notranslate"><span class="pre">quaternion_inverse()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_matrix"><code class="docutils literal notranslate"><span class="pre">quaternion_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_multiply"><code class="docutils literal notranslate"><span class="pre">quaternion_multiply()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_real"><code class="docutils literal notranslate"><span class="pre">quaternion_real()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.quaternion_slerp"><code class="docutils literal notranslate"><span class="pre">quaternion_slerp()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.random_quaternion"><code class="docutils literal notranslate"><span class="pre">random_quaternion()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.random_rotation_matrix"><code class="docutils literal notranslate"><span class="pre">random_rotation_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.random_vector"><code class="docutils literal notranslate"><span class="pre">random_vector()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.reflection_from_matrix"><code class="docutils literal notranslate"><span class="pre">reflection_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.reflection_matrix"><code class="docutils literal notranslate"><span class="pre">reflection_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.rotation_from_matrix"><code class="docutils literal notranslate"><span class="pre">rotation_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.rotation_matrix"><code class="docutils literal notranslate"><span class="pre">rotation_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.scale_from_matrix"><code class="docutils literal notranslate"><span class="pre">scale_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.scale_matrix"><code class="docutils literal notranslate"><span class="pre">scale_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.shear_from_matrix"><code class="docutils literal notranslate"><span class="pre">shear_from_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.shear_matrix"><code class="docutils literal notranslate"><span class="pre">shear_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.superimposition_matrix"><code class="docutils literal notranslate"><span class="pre">superimposition_matrix()</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.translation_from_matrix"><code class="docutils literal notranslate"><span class="pre">translation_from_matrix()</span></code></a></li>
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<li class="toctree-l3"><a class="reference internal" href="#espressopp.external.transformations.vector_product"><code class="docutils literal notranslate"><span class="pre">vector_product()</span></code></a></li>
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<section id="espressopp-external-transformations">
<span id="module-espressopp.external.transformations"></span><h1>espressopp.external.transformations<a class="headerlink" href="#espressopp-external-transformations" title="Link to this heading"></a></h1>
<p>Homogeneous Transformation Matrices and Quaternions.</p>
<p>A library for calculating 4x4 matrices for translating, rotating, reflecting,
scaling, shearing, projecting, orthogonalizing, and superimposing arrays of
3D homogeneous coordinates as well as for converting between rotation matrices,
Euler angles, and quaternions. Also includes an Arcball control object and
functions to decompose transformation matrices.</p>
<dl class="field-list simple">
<dt class="field-odd">Authors<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="http://www.lfd.uci.edu/~gohlke/">Christoph Gohlke</a>,
Laboratory for Fluorescence Dynamics, University of California, Irvine</p>
</dd>
<dt class="field-even">Version<span class="colon">:</span></dt>
<dd class="field-even"><p>2011.01.25</p>
</dd>
</dl>
<p><strong>Requirements</strong></p>
<ul class="simple">
<li><p><a class="reference external" href="http://www.python.org">Python 2.6 or 3.1</a></p></li>
<li><p><a class="reference external" href="http://numpy.scipy.org">Numpy 1.5</a></p></li>
<li><p><a class="reference external" href="http://www.lfd.uci.edu/~gohlke/">transformations.c 2010.04.10</a>
(optional implementation of some functions in C)</p></li>
</ul>
<p><strong>Notes</strong></p>
<p>The API is not stable yet and is expected to change between revisions.</p>
<p>This Python code is not optimized for speed. Refer to the transformations.c
module for a faster implementation of some functions.</p>
<p>Documentation in HTML format can be generated with epydoc.</p>
<p>Matrices (M) can be inverted using numpy.linalg.inv(M), concatenated using
numpy.dot(M0, M1), or used to transform homogeneous coordinates (v) using
numpy.dot(M, v) for shape (4, *) “point of arrays”, respectively
numpy.dot(v, M.T) for shape (*, 4) “array of points”.</p>
<p>Use the transpose of transformation matrices for OpenGL glMultMatrixd().</p>
<p>Calculations are carried out with numpy.float64 precision.</p>
<p>Vector, point, quaternion, and matrix function arguments are expected to be
“array like”, i.e. tuple, list, or numpy arrays.</p>
<p>Return types are numpy arrays unless specified otherwise.</p>
<p>Angles are in radians unless specified otherwise.</p>
<p>Quaternions w+ix+jy+kz are represented as [w, x, y, z].</p>
<p>A triple of Euler angles can be applied/interpreted in 24 ways, which can
be specified using a 4 character string or encoded 4-tuple:</p>
<blockquote>
<div><p><em>Axes 4-string</em>: e.g. ‘sxyz’ or ‘ryxy’</p>
<ul class="simple">
<li><p>first character : rotations are applied to ‘s’tatic or ‘r’otating frame</p></li>
<li><p>remaining characters : successive rotation axis ‘x’, ‘y’, or ‘z’</p></li>
</ul>
<p><em>Axes 4-tuple</em>: e.g. (0, 0, 0, 0) or (1, 1, 1, 1)</p>
<ul class="simple">
<li><p>inner axis: code of axis (‘x’:0, ‘y’:1, ‘z’:2) of rightmost matrix.</p></li>
<li><p>parity : even (0) if inner axis ‘x’ is followed by ‘y’, ‘y’ is followed
by ‘z’, or ‘z’ is followed by ‘x’. Otherwise odd (1).</p></li>
<li><p>repetition : first and last axis are same (1) or different (0).</p></li>
<li><p>frame : rotations are applied to static (0) or rotating (1) frame.</p></li>
</ul>
</div></blockquote>
<p><strong>References</strong></p>
<ol class="arabic simple">
<li><p>Matrices and transformations. Ronald Goldman.
In “Graphics Gems I”, pp 472-475. Morgan Kaufmann, 1990.</p></li>
<li><p>More matrices and transformations: shear and pseudo-perspective.
Ronald Goldman. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.</p></li>
<li><p>Decomposing a matrix into simple transformations. Spencer Thomas.
In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.</p></li>
<li><p>Recovering the data from the transformation matrix. Ronald Goldman.
In “Graphics Gems II”, pp 324-331. Morgan Kaufmann, 1991.</p></li>
<li><p>Euler angle conversion. Ken Shoemake.
In “Graphics Gems IV”, pp 222-229. Morgan Kaufmann, 1994.</p></li>
<li><p>Arcball rotation control. Ken Shoemake.
In “Graphics Gems IV”, pp 175-192. Morgan Kaufmann, 1994.</p></li>
<li><p>Representing attitude: Euler angles, unit quaternions, and rotation
vectors. James Diebel. 2006.</p></li>
<li><p>A discussion of the solution for the best rotation to relate two sets
of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.</p></li>
<li><p>Closed-form solution of absolute orientation using unit quaternions.
BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642.</p></li>
<li><p>Quaternions. Ken Shoemake.
<a class="reference external" href="http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf">http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf</a></p></li>
<li><p>From quaternion to matrix and back. JMP van Waveren. 2005.
<a class="reference external" href="http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm">http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm</a></p></li>
<li><p>Uniform random rotations. Ken Shoemake.
In “Graphics Gems III”, pp 124-132. Morgan Kaufmann, 1992.</p></li>
<li><p>Quaternion in molecular modeling. CFF Karney.
J Mol Graph Mod, 25(5):595-604</p></li>
<li><p>New method for extracting the quaternion from a rotation matrix.
Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087.</p></li>
</ol>
<p>Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.123</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.234</span><span class="p">,</span> <span class="mf">2.345</span>
<span class="gp">>>> </span><span class="n">origin</span><span class="p">,</span> <span class="n">xaxis</span><span class="p">,</span> <span class="n">yaxis</span><span class="p">,</span> <span class="n">zaxis</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">I</span> <span class="o">=</span> <span class="n">identity_matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">Rx</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">xaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Ry</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">yaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Rz</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="n">zaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">concatenate_matrices</span><span class="p">(</span><span class="n">Rx</span><span class="p">,</span> <span class="n">Ry</span><span class="p">,</span> <span class="n">Rz</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">euler</span> <span class="o">=</span> <span class="n">euler_from_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="s1">'rxyz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">([</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">],</span> <span class="n">euler</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">Re</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">gamma</span><span class="p">,</span> <span class="s1">'rxyz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">Re</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">al</span><span class="p">,</span> <span class="n">be</span><span class="p">,</span> <span class="n">ga</span> <span class="o">=</span> <span class="n">euler_from_matrix</span><span class="p">(</span><span class="n">Re</span><span class="p">,</span> <span class="s1">'rxyz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">Re</span><span class="p">,</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">al</span><span class="p">,</span> <span class="n">be</span><span class="p">,</span> <span class="n">ga</span><span class="p">,</span> <span class="s1">'rxyz'</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">qx</span> <span class="o">=</span> <span class="n">quaternion_about_axis</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">xaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">qy</span> <span class="o">=</span> <span class="n">quaternion_about_axis</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">yaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">qz</span> <span class="o">=</span> <span class="n">quaternion_about_axis</span><span class="p">(</span><span class="n">gamma</span><span class="p">,</span> <span class="n">zaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_multiply</span><span class="p">(</span><span class="n">qx</span><span class="p">,</span> <span class="n">qy</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_multiply</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">qz</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">Rq</span> <span class="o">=</span> <span class="n">quaternion_matrix</span><span class="p">(</span><span class="n">q</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">Rq</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="mf">1.23</span><span class="p">,</span> <span class="n">origin</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">T</span> <span class="o">=</span> <span class="n">translation_matrix</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">Z</span> <span class="o">=</span> <span class="n">shear_matrix</span><span class="p">(</span><span class="n">beta</span><span class="p">,</span> <span class="n">xaxis</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">zaxis</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">random_rotation_matrix</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">concatenate_matrices</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">R</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span> <span class="o">=</span> <span class="n">decompose_matrix</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">scale</span><span class="p">,</span> <span class="mf">1.23</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">trans</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">shear</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">tan</span><span class="p">(</span><span class="n">beta</span><span class="p">),</span> <span class="mi">0</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="s1">'sxyz'</span><span class="p">,</span> <span class="o">*</span><span class="n">angles</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">M1</span> <span class="o">=</span> <span class="n">compose_matrix</span><span class="p">(</span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">M1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span> <span class="o">=</span> <span class="n">random_vector</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span> <span class="n">random_vector</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle_between_vectors</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">),</span> <span class="n">vector_product</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v2</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">M</span><span class="p">[:</span><span class="mi">3</span><span class="p">,:</span><span class="mi">3</span><span class="p">]</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">unit_vector</span><span class="p">(</span><span class="n">v1</span><span class="p">),</span> <span class="n">unit_vector</span><span class="p">(</span><span class="n">v2</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</section>
<dl class="py class">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball">
<em class="property"><span class="k"><span class="pre">class</span></span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">Arcball</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">initial</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball" title="Link to this definition"></a></dt>
<dd><p>Virtual Trackball Control.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ball</span> <span class="o">=</span> <span class="n">Arcball</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">ball</span> <span class="o">=</span> <span class="n">Arcball</span><span class="p">(</span><span class="n">initial</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">place</span><span class="p">([</span><span class="mi">320</span><span class="p">,</span> <span class="mi">320</span><span class="p">],</span> <span class="mi">320</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">down</span><span class="p">([</span><span class="mi">500</span><span class="p">,</span> <span class="mi">250</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">drag</span><span class="p">([</span><span class="mi">475</span><span class="p">,</span> <span class="mi">275</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">ball</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">R</span><span class="p">),</span> <span class="mf">3.90583455</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">ball</span> <span class="o">=</span> <span class="n">Arcball</span><span class="p">(</span><span class="n">initial</span><span class="o">=</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">place</span><span class="p">([</span><span class="mi">320</span><span class="p">,</span> <span class="mi">320</span><span class="p">],</span> <span class="mi">320</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">setaxes</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">setconstrain</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">down</span><span class="p">([</span><span class="mi">400</span><span class="p">,</span> <span class="mi">200</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">drag</span><span class="p">([</span><span class="mi">200</span><span class="p">,</span> <span class="mi">400</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">ball</span><span class="o">.</span><span class="n">matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">R</span><span class="p">),</span> <span class="mf">0.2055924</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">ball</span><span class="o">.</span><span class="n">next</span><span class="p">()</span>
</pre></div>
</div>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.down">
<span class="sig-name descname"><span class="pre">down</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.down" title="Link to this definition"></a></dt>
<dd><p>Set initial cursor window coordinates and pick constrain-axis.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.drag">
<span class="sig-name descname"><span class="pre">drag</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.drag" title="Link to this definition"></a></dt>
<dd><p>Update current cursor window coordinates.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.getconstrain">
<span class="sig-name descname"><span class="pre">getconstrain</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.getconstrain" title="Link to this definition"></a></dt>
<dd><p>Return state of constrain to axis mode.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.matrix">
<span class="sig-name descname"><span class="pre">matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.matrix" title="Link to this definition"></a></dt>
<dd><p>Return homogeneous rotation matrix.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.next">
<span class="sig-name descname"><span class="pre">next</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">acceleration</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.next" title="Link to this definition"></a></dt>
<dd><p>Continue rotation in direction of last drag.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.place">
<span class="sig-name descname"><span class="pre">place</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">center</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">radius</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.place" title="Link to this definition"></a></dt>
<dd><p>Place Arcball, e.g. when window size changes.</p>
<dl class="simple">
<dt>center<span class="classifier">sequence[2]</span></dt><dd><p>Window coordinates of trackball center.</p>
</dd>
<dt>radius<span class="classifier">float</span></dt><dd><p>Radius of trackball in window coordinates.</p>
</dd>
</dl>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.setaxes">
<span class="sig-name descname"><span class="pre">setaxes</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">axes</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.setaxes" title="Link to this definition"></a></dt>
<dd><p>Set axes to constrain rotations.</p>
</dd></dl>
<dl class="py method">
<dt class="sig sig-object py" id="espressopp.external.transformations.Arcball.setconstrain">
<span class="sig-name descname"><span class="pre">setconstrain</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">constrain</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.Arcball.setconstrain" title="Link to this definition"></a></dt>
<dd><p>Set state of constrain to axis mode.</p>
</dd></dl>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.angle_between_vectors">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">angle_between_vectors</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">v0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">v1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">directed</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.angle_between_vectors" title="Link to this definition"></a></dt>
<dd><p>Return angle between vectors.</p>
<p>If directed is False, the input vectors are interpreted as undirected axes,
i.e. the maximum angle is pi/2.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">angle_between_vectors</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">angle_between_vectors</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span> <span class="n">directed</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">angle_between_vectors</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.5708</span><span class="p">,</span> <span class="mf">1.5708</span><span class="p">,</span> <span class="mf">0.95532</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">angle_between_vectors</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.5708</span><span class="p">,</span> <span class="mf">1.5708</span><span class="p">,</span> <span class="mf">1.5708</span><span class="p">,</span> <span class="mf">0.95532</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.arcball_constrain_to_axis">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">arcball_constrain_to_axis</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.arcball_constrain_to_axis" title="Link to this definition"></a></dt>
<dd><p>Return sphere point perpendicular to axis.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.arcball_map_to_sphere">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">arcball_map_to_sphere</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">center</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">radius</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.arcball_map_to_sphere" title="Link to this definition"></a></dt>
<dd><p>Return unit sphere coordinates from window coordinates.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.arcball_nearest_axis">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">arcball_nearest_axis</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.arcball_nearest_axis" title="Link to this definition"></a></dt>
<dd><p>Return axis, which arc is nearest to point.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.clip_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">clip_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">left</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">right</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">bottom</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">top</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">near</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">far</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">perspective</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.clip_matrix" title="Link to this definition"></a></dt>
<dd><p>Return matrix to obtain normalized device coordinates from frustrum.</p>
<p>The frustrum bounds are axis-aligned along x (left, right),
y (bottom, top) and z (near, far).</p>
<p>Normalized device coordinates are in range [-1, 1] if coordinates are
inside the frustrum.</p>
<p>If perspective is True the frustrum is a truncated pyramid with the
perspective point at origin and direction along z axis, otherwise an
orthographic canonical view volume (a box).</p>
<p>Homogeneous coordinates transformed by the perspective clip matrix
need to be dehomogenized (devided by w coordinate).</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">frustrum</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">frustrum</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">frustrum</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">+=</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">frustrum</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span> <span class="o">+=</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">4</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">clip_matrix</span><span class="p">(</span><span class="n">perspective</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">*</span><span class="n">frustrum</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">[</span><span class="n">frustrum</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="mf">1.0</span><span class="p">])</span>
<span class="go">array([-1., -1., -1., 1.])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">[</span><span class="n">frustrum</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">5</span><span class="p">],</span> <span class="mf">1.0</span><span class="p">])</span>
<span class="go">array([ 1., 1., 1., 1.])</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">clip_matrix</span><span class="p">(</span><span class="n">perspective</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">*</span><span class="n">frustrum</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">[</span><span class="n">frustrum</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="mf">1.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">/</span> <span class="n">v</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
<span class="go">array([-1., -1., -1., 1.])</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="p">[</span><span class="n">frustrum</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="n">frustrum</span><span class="p">[</span><span class="mi">4</span><span class="p">],</span> <span class="mf">1.0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">/</span> <span class="n">v</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span>
<span class="go">array([ 1., 1., -1., 1.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.compose_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">compose_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">scale</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">shear</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">angles</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">translate</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">perspective</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.compose_matrix" title="Link to this definition"></a></dt>
<dd><p>Return transformation matrix from sequence of transformations.</p>
<p>This is the inverse of the decompose_matrix function.</p>
<dl class="simple">
<dt>Sequence of transformations:</dt><dd><p>scale : vector of 3 scaling factors
shear : list of shear factors for x-y, x-z, y-z axes
angles : list of Euler angles about static x, y, z axes
translate : translation vector along x, y, z axes
perspective : perspective partition of matrix</p>
</dd>
</dl>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">scale</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">shear</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">angles</span> <span class="o">=</span> <span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">trans</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">persp</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">M0</span> <span class="o">=</span> <span class="n">compose_matrix</span><span class="p">(</span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">decompose_matrix</span><span class="p">(</span><span class="n">M0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M1</span> <span class="o">=</span> <span class="n">compose_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">M0</span><span class="p">,</span> <span class="n">M1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.concatenate_matrices">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">concatenate_matrices</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">matrices</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.concatenate_matrices" title="Link to this definition"></a></dt>
<dd><p>Return concatenation of series of transformation matrices.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">16</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">concatenate_matrices</span><span class="p">(</span><span class="n">M</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">T</span><span class="p">),</span> <span class="n">concatenate_matrices</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.decompose_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">decompose_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.decompose_matrix" title="Link to this definition"></a></dt>
<dd><p>Return sequence of transformations from transformation matrix.</p>
<dl class="simple">
<dt>matrix<span class="classifier">array_like</span></dt><dd><p>Non-degenerative homogeneous transformation matrix</p>
</dd>
<dt>Return tuple of:</dt><dd><p>scale : vector of 3 scaling factors
shear : list of shear factors for x-y, x-z, y-z axes
angles : list of Euler angles about static x, y, z axes
translate : translation vector along x, y, z axes
perspective : perspective partition of matrix</p>
</dd>
</dl>
<p>Raise ValueError if matrix is of wrong type or degenerative.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">T0</span> <span class="o">=</span> <span class="n">translation_matrix</span><span class="p">((</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span> <span class="o">=</span> <span class="n">decompose_matrix</span><span class="p">(</span><span class="n">T0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">T1</span> <span class="o">=</span> <span class="n">translation_matrix</span><span class="p">(</span><span class="n">trans</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">T0</span><span class="p">,</span> <span class="n">T1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="mf">0.123</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span> <span class="o">=</span> <span class="n">decompose_matrix</span><span class="p">(</span><span class="n">S</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">scale</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="go">0.123</span>
<span class="gp">>>> </span><span class="n">R0</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">scale</span><span class="p">,</span> <span class="n">shear</span><span class="p">,</span> <span class="n">angles</span><span class="p">,</span> <span class="n">trans</span><span class="p">,</span> <span class="n">persp</span> <span class="o">=</span> <span class="n">decompose_matrix</span><span class="p">(</span><span class="n">R0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R1</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">angles</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.euler_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">euler_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sxyz'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.euler_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return Euler angles from rotation matrix for specified axis sequence.</p>
<p>axes : One of 24 axis sequences as string or encoded tuple</p>
<p>Note that many Euler angle triplets can describe one matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">R0</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="s1">'syxz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">al</span><span class="p">,</span> <span class="n">be</span><span class="p">,</span> <span class="n">ga</span> <span class="o">=</span> <span class="n">euler_from_matrix</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="s1">'syxz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R1</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">al</span><span class="p">,</span> <span class="n">be</span><span class="p">,</span> <span class="n">ga</span><span class="p">,</span> <span class="s1">'syxz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">angles</span> <span class="o">=</span> <span class="p">(</span><span class="mf">4.0</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">axes</span> <span class="ow">in</span> <span class="n">_AXES2TUPLE</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">R0</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="o">*</span><span class="n">angles</span><span class="p">)</span>
<span class="gp">... </span> <span class="n">R1</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">axes</span><span class="o">=</span><span class="n">axes</span><span class="p">,</span> <span class="o">*</span><span class="n">euler_from_matrix</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">axes</span><span class="p">))</span>
<span class="gp">... </span> <span class="k">if</span> <span class="ow">not</span> <span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">):</span> <span class="nb">print</span><span class="p">(</span><span class="n">axes</span><span class="p">,</span> <span class="s2">"failed"</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.euler_from_quaternion">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">euler_from_quaternion</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sxyz'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.euler_from_quaternion" title="Link to this definition"></a></dt>
<dd><p>Return Euler angles from quaternion for specified axis sequence.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">angles</span> <span class="o">=</span> <span class="n">euler_from_quaternion</span><span class="p">([</span><span class="mf">0.99810947</span><span class="p">,</span> <span class="mf">0.06146124</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">angles</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.123</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.euler_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">euler_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ai</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">aj</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ak</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sxyz'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.euler_matrix" title="Link to this definition"></a></dt>
<dd><p>Return homogeneous rotation matrix from Euler angles and axis sequence.</p>
<p>ai, aj, ak : Euler’s roll, pitch and yaw angles
axes : One of 24 axis sequences as string or encoded tuple</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="s1">'syxz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="o">-</span><span class="mf">1.34786452</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">R</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="o">-</span><span class="mf">0.383436184</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">ai</span><span class="p">,</span> <span class="n">aj</span><span class="p">,</span> <span class="n">ak</span> <span class="o">=</span> <span class="p">(</span><span class="mf">4.0</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">axes</span> <span class="ow">in</span> <span class="n">_AXES2TUPLE</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">R</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">ai</span><span class="p">,</span> <span class="n">aj</span><span class="p">,</span> <span class="n">ak</span><span class="p">,</span> <span class="n">axes</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">axes</span> <span class="ow">in</span> <span class="n">_TUPLE2AXES</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span>
<span class="gp">... </span> <span class="n">R</span> <span class="o">=</span> <span class="n">euler_matrix</span><span class="p">(</span><span class="n">ai</span><span class="p">,</span> <span class="n">aj</span><span class="p">,</span> <span class="n">ak</span><span class="p">,</span> <span class="n">axes</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.identity_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">identity_matrix</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.identity_matrix" title="Link to this definition"></a></dt>
<dd><p>Return 4x4 identity/unit matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">I</span> <span class="o">=</span> <span class="n">identity_matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">I</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">I</span><span class="p">),</span> <span class="n">numpy</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">I</span><span class="p">)</span>
<span class="go">(4.0, 4.0)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">numpy</span><span class="o">.</span><span class="n">float64</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.inverse_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">inverse_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.inverse_matrix" title="Link to this definition"></a></dt>
<dd><p>Return inverse of square transformation matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">M0</span> <span class="o">=</span> <span class="n">random_rotation_matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">M1</span> <span class="o">=</span> <span class="n">inverse_matrix</span><span class="p">(</span><span class="n">M0</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M1</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">M0</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="k">for</span> <span class="n">size</span> <span class="ow">in</span> <span class="n">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">7</span><span class="p">):</span>
<span class="gp">... </span> <span class="n">M0</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">size</span><span class="p">,</span> <span class="n">size</span><span class="p">)</span>
<span class="gp">... </span> <span class="n">M1</span> <span class="o">=</span> <span class="n">inverse_matrix</span><span class="p">(</span><span class="n">M0</span><span class="p">)</span>
<span class="gp">... </span> <span class="k">if</span> <span class="ow">not</span> <span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M1</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">M0</span><span class="p">)):</span> <span class="nb">print</span><span class="p">(</span><span class="n">size</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.is_same_transform">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">is_same_transform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">matrix1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.is_same_transform" title="Link to this definition"></a></dt>
<dd><p>Return True if two matrices perform same transformation.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">random_rotation_matrix</span><span class="p">())</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.orthogonalization_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">orthogonalization_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lengths</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">angles</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.orthogonalization_matrix" title="Link to this definition"></a></dt>
<dd><p>Return orthogonalization matrix for crystallographic cell coordinates.</p>
<p>Angles are expected in degrees.</p>
<p>The de-orthogonalization matrix is the inverse.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">O</span> <span class="o">=</span> <span class="n">orthogonalization_matrix</span><span class="p">((</span><span class="mf">10.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">,</span> <span class="mf">10.</span><span class="p">),</span> <span class="p">(</span><span class="mf">90.</span><span class="p">,</span> <span class="mf">90.</span><span class="p">,</span> <span class="mf">90.</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">O</span><span class="p">[:</span><span class="mi">3</span><span class="p">,</span> <span class="p">:</span><span class="mi">3</span><span class="p">],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="nb">float</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">O</span> <span class="o">=</span> <span class="n">orthogonalization_matrix</span><span class="p">([</span><span class="mf">9.8</span><span class="p">,</span> <span class="mf">12.0</span><span class="p">,</span> <span class="mf">15.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">87.2</span><span class="p">,</span> <span class="mf">80.7</span><span class="p">,</span> <span class="mf">69.7</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">O</span><span class="p">),</span> <span class="mf">43.063229</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.projection_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">projection_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pseudo</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.projection_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return projection plane and perspective point from projection matrix.</p>
<p>Return values are same as arguments for projection_matrix function:
point, normal, direction, perspective, and pseudo.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">point</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">normal</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">direct</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">persp</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">P0</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">projection_from_matrix</span><span class="p">(</span><span class="n">P0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P1</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">P1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">P0</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">direct</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">projection_from_matrix</span><span class="p">(</span><span class="n">P0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P1</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">P1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">P0</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">perspective</span><span class="o">=</span><span class="n">persp</span><span class="p">,</span> <span class="n">pseudo</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">projection_from_matrix</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">pseudo</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P1</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">P1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">P0</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">perspective</span><span class="o">=</span><span class="n">persp</span><span class="p">,</span> <span class="n">pseudo</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">projection_from_matrix</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">pseudo</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P1</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="o">*</span><span class="n">result</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">P1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.projection_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">projection_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">normal</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">direction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">perspective</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">pseudo</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.projection_matrix" title="Link to this definition"></a></dt>
<dd><p>Return matrix to project onto plane defined by point and normal.</p>
<p>Using either perspective point, projection direction, or none of both.</p>
<p>If pseudo is True, perspective projections will preserve relative depth
such that Perspective = dot(Orthogonal, PseudoPerspective).</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">P</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">((</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">:,</span> <span class="mi">1</span><span class="p">:],</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">)[</span><span class="mi">1</span><span class="p">:,</span> <span class="mi">1</span><span class="p">:])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">point</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">normal</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">direct</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">persp</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">P0</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P1</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">direction</span><span class="o">=</span><span class="n">direct</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P2</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">perspective</span><span class="o">=</span><span class="n">persp</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">P3</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">,</span> <span class="n">perspective</span><span class="o">=</span><span class="n">persp</span><span class="p">,</span> <span class="n">pseudo</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">P2</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">P0</span><span class="p">,</span> <span class="n">P3</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">P</span> <span class="o">=</span> <span class="n">projection_matrix</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="mf">20.0</span>
<span class="gp">>>> </span><span class="n">v0</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">v0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v1</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">v0</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">3.0</span><span class="o">-</span><span class="n">v1</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_about_axis">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_about_axis</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">angle</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axis</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_about_axis" title="Link to this definition"></a></dt>
<dd><p>Return quaternion for rotation about axis.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_about_axis</span><span class="p">(</span><span class="mf">0.123</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.99810947</span><span class="p">,</span> <span class="mf">0.06146124</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_conjugate">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_conjugate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_conjugate" title="Link to this definition"></a></dt>
<dd><p>Return conjugate of quaternion.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q0</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">=</span> <span class="n">quaternion_conjugate</span><span class="p">(</span><span class="n">q0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">q1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">==</span> <span class="n">q0</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="nb">all</span><span class="p">(</span><span class="n">q1</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span> <span class="o">==</span> <span class="o">-</span><span class="n">q0</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_from_euler">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_from_euler</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ai</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">aj</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ak</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">axes</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'sxyz'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_from_euler" title="Link to this definition"></a></dt>
<dd><p>Return quaternion from Euler angles and axis sequence.</p>
<p>ai, aj, ak : Euler’s roll, pitch and yaw angles
axes : One of 24 axis sequences as string or encoded tuple</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_euler</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="s1">'ryxz'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.435953</span><span class="p">,</span> <span class="mf">0.310622</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.718287</span><span class="p">,</span> <span class="mf">0.444435</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">isprecise</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return quaternion from rotation matrix.</p>
<p>If isprecise=True, the input matrix is assumed to be a precise rotation
matrix and a faster algorithm is used.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">identity_matrix</span><span class="p">(),</span> <span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">diag</span><span class="p">([</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span> <span class="ow">or</span> <span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="mf">0.123</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.9981095</span><span class="p">,</span> <span class="mf">0.0164262</span><span class="p">,</span> <span class="mf">0.0328524</span><span class="p">,</span> <span class="mf">0.0492786</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mf">0.545</span><span class="p">,</span> <span class="mf">0.797</span><span class="p">,</span> <span class="mf">0.260</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.733</span><span class="p">,</span> <span class="mf">0.603</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.313</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="o">-</span><span class="mf">0.407</span><span class="p">,</span> <span class="mf">0.021</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.913</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.19069</span><span class="p">,</span> <span class="mf">0.43736</span><span class="p">,</span> <span class="mf">0.87485</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.083611</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.395</span><span class="p">,</span> <span class="mf">0.362</span><span class="p">,</span> <span class="mf">0.843</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.626</span><span class="p">,</span> <span class="mf">0.796</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.056</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="o">-</span><span class="mf">0.677</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.498</span><span class="p">,</span> <span class="mf">0.529</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mf">0.82336615</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.13610694</span><span class="p">,</span> <span class="mf">0.46344705</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.29792603</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">random_rotation_matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_from_matrix</span><span class="p">(</span><span class="n">R</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">quaternion_matrix</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_imag">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_imag</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_imag" title="Link to this definition"></a></dt>
<dd><p>Return imaginary part of quaternion.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">quaternion_imag</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
<span class="go">[0.0, 1.0, 2.0]</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_inverse">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_inverse</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_inverse" title="Link to this definition"></a></dt>
<dd><p>Return inverse of quaternion.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q0</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">=</span> <span class="n">quaternion_inverse</span><span class="p">(</span><span class="n">q0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">quaternion_multiply</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">),</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_matrix" title="Link to this definition"></a></dt>
<dd><p>Return homogeneous rotation matrix from quaternion.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">quaternion_matrix</span><span class="p">([</span><span class="mf">0.99810947</span><span class="p">,</span> <span class="mf">0.06146124</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="mf">0.123</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">quaternion_matrix</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">identity_matrix</span><span class="p">())</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">quaternion_matrix</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">diag</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_multiply">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_multiply</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">quaternion0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_multiply" title="Link to this definition"></a></dt>
<dd><p>Return multiplication of two quaternions.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_multiply</span><span class="p">([</span><span class="mi">4</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="p">[</span><span class="mi">28</span><span class="p">,</span> <span class="o">-</span><span class="mi">44</span><span class="p">,</span> <span class="o">-</span><span class="mi">14</span><span class="p">,</span> <span class="mi">48</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_real">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_real</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quaternion</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_real" title="Link to this definition"></a></dt>
<dd><p>Return real part of quaternion.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">quaternion_real</span><span class="p">([</span><span class="mf">3.0</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">])</span>
<span class="go">3.0</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.quaternion_slerp">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">quaternion_slerp</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">quat0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">quat1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">fraction</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">spin</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">shortestpath</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.quaternion_slerp" title="Link to this definition"></a></dt>
<dd><p>Return spherical linear interpolation between two quaternions.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q0</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">q1</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_slerp</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">q0</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_slerp</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="n">q1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">quaternion_slerp</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">angle</span> <span class="o">=</span> <span class="n">math</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">))</span> <span class="o">/</span> <span class="n">angle</span><span class="p">)</span> <span class="ow">or</span> <span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="mf">2.0</span><span class="p">,</span> <span class="n">math</span><span class="o">.</span><span class="n">acos</span><span class="p">(</span><span class="o">-</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">q0</span><span class="p">,</span> <span class="n">q1</span><span class="p">))</span> <span class="o">/</span> <span class="n">angle</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.random_quaternion">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">random_quaternion</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rand</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.random_quaternion" title="Link to this definition"></a></dt>
<dd><p>Return uniform random unit quaternion.</p>
<dl class="simple">
<dt>rand: array like or None</dt><dd><p>Three independent random variables that are uniformly distributed
between 0 and 1.</p>
</dd>
</dl>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">vector_norm</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">q</span> <span class="o">=</span> <span class="n">random_quaternion</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">shape</span><span class="p">),</span> <span class="n">q</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">==</span><span class="mi">4</span>
<span class="go">(1, True)</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.random_rotation_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">random_rotation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">rand</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.random_rotation_matrix" title="Link to this definition"></a></dt>
<dd><p>Return uniform random rotation matrix.</p>
<dl class="simple">
<dt>rnd: array like</dt><dd><p>Three independent random variables that are uniformly distributed
between 0 and 1 for each returned quaternion.</p>
</dd>
</dl>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">random_rotation_matrix</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">R</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">R</span><span class="p">),</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.random_vector">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">random_vector</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">size</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.random_vector" title="Link to this definition"></a></dt>
<dd><p>Return array of random doubles in the half-open interval [0.0, 1.0).</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">random_vector</span><span class="p">(</span><span class="mi">10000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">v</span> <span class="o">>=</span> <span class="mf">0.0</span><span class="p">)</span> <span class="ow">and</span> <span class="n">numpy</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">v</span> <span class="o"><</span> <span class="mf">1.0</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="n">random_vector</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">random_vector</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">v0</span> <span class="o">==</span> <span class="n">v1</span><span class="p">)</span>
<span class="go">False</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.reflection_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">reflection_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.reflection_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return mirror plane point and normal vector from reflection matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">M0</span> <span class="o">=</span> <span class="n">reflection_matrix</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span> <span class="o">=</span> <span class="n">reflection_from_matrix</span><span class="p">(</span><span class="n">M0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M1</span> <span class="o">=</span> <span class="n">reflection_matrix</span><span class="p">(</span><span class="n">point</span><span class="p">,</span> <span class="n">normal</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">M0</span><span class="p">,</span> <span class="n">M1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.reflection_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">reflection_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">point</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">normal</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.reflection_matrix" title="Link to this definition"></a></dt>
<dd><p>Return matrix to mirror at plane defined by point and normal vector.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">v0</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">reflection_matrix</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="mf">2.</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">R</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v0</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">v0</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v2</span> <span class="o">=</span> <span class="n">v0</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">v2</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span> <span class="o">+=</span> <span class="n">v1</span>
<span class="gp">>>> </span><span class="n">v3</span> <span class="o">=</span> <span class="n">v0</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">v2</span><span class="p">[:</span><span class="mi">3</span><span class="p">]</span> <span class="o">-=</span> <span class="n">v1</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v2</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="n">v3</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.rotation_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">rotation_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.rotation_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return rotation angle and axis from rotation matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">angle</span> <span class="o">=</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">direc</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">point</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">R0</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">angle</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span> <span class="o">=</span> <span class="n">rotation_from_matrix</span><span class="p">(</span><span class="n">R0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R1</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.rotation_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">rotation_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">angle</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">direction</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">point</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.rotation_matrix" title="Link to this definition"></a></dt>
<dd><p>Return matrix to rotate about axis defined by point and direction.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">R</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mf">2.0</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">R</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]),</span> <span class="p">[</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">])</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">angle</span> <span class="o">=</span> <span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">direc</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">point</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">R0</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R1</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">R0</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">R1</span> <span class="o">=</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="o">-</span><span class="n">angle</span><span class="p">,</span> <span class="o">-</span><span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">R0</span><span class="p">,</span> <span class="n">R1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">I</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">rotation_matrix</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="mi">2</span><span class="p">,</span> <span class="n">direc</span><span class="p">))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="mf">2.</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">trace</span><span class="p">(</span><span class="n">rotation_matrix</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">direc</span><span class="p">,</span> <span class="n">point</span><span class="p">)))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.scale_from_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">scale_from_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">matrix</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.scale_from_matrix" title="Link to this definition"></a></dt>
<dd><p>Return scaling factor, origin and direction from scaling matrix.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">factor</span> <span class="o">=</span> <span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">-</span> <span class="mi">5</span>
<span class="gp">>>> </span><span class="n">origin</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">direct</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="gp">>>> </span><span class="n">S0</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">direction</span> <span class="o">=</span> <span class="n">scale_from_matrix</span><span class="p">(</span><span class="n">S0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">S1</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">direction</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">S0</span><span class="p">,</span> <span class="n">S1</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">S0</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">direct</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">direction</span> <span class="o">=</span> <span class="n">scale_from_matrix</span><span class="p">(</span><span class="n">S0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">S1</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="n">factor</span><span class="p">,</span> <span class="n">origin</span><span class="p">,</span> <span class="n">direction</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">is_same_transform</span><span class="p">(</span><span class="n">S0</span><span class="p">,</span> <span class="n">S1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="espressopp.external.transformations.scale_matrix">
<span class="sig-prename descclassname"><span class="pre">espressopp.external.transformations.</span></span><span class="sig-name descname"><span class="pre">scale_matrix</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">factor</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">origin</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">direction</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#espressopp.external.transformations.scale_matrix" title="Link to this definition"></a></dt>
<dd><p>Return matrix to scale by factor around origin in direction.</p>
<p>Use factor -1 for point symmetry.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">*</span> <span class="mf">20.0</span>
<span class="gp">>>> </span><span class="n">v</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
<span class="gp">>>> </span><span class="n">S</span> <span class="o">=</span> <span class="n">scale_matrix</span><span class="p">(</span><span class="o">-</span><span class="mf">1.234</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">S</span><span class="p">,</span> <span class="n">v</span><span class="p">)[:</span><span class="mi">3</span><span class="p">],</span> <span class="o">-</span><span class="mf">1.234</span><span class="o">*</span><span class="n">v</span><span class="p">[:</span><span class="mi">3</span><span class="p">])</span>
<span class="go">True</span>