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<div class="section" id="module-transformations">
<span id="the-transformations-module"></span><span id="transformations"></span><h1>The Transformations Module<a class="headerlink" href="#module-transformations" title="Permalink to this headline">¶</a></h1>
<p>Homogeneous Transformation Matrices and Quaternions.</p>
<p><strong>Project Name:</strong> MakeHuman</p>
<p><strong>Product Home Page:</strong> <a class="reference external" href="http://www.makehuman.org/">http://www.makehuman.org/</a></p>
<p><strong>Code Home Page:</strong> <a class="reference external" href="https://bitbucket.org/MakeHuman/makehuman/">https://bitbucket.org/MakeHuman/makehuman/</a></p>
<dl class="docutils">
<dt><strong>Authors:</strong> <a class="reference external" href="http://www.lfd.uci.edu/~gohlke/">Christoph Gohlke</a>,</dt>
<dd>Laboratory for Fluorescence Dynamics, University of California, Irvine</dd>
</dl>
<p><strong>Copyright(c):</strong> MakeHuman Team 2001-2017</p>
<p><strong>Licensing:</strong> AGPL3</p>
<blockquote>
<div><p>This file is part of MakeHuman (www.makehuman.org).</p>
<p>This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Affero General Public License as
published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.</p>
<p>This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Affero General Public License for more details.</p>
<p>You should have received a copy of the GNU Affero General Public License
along with this program. If not, see <<a class="reference external" href="http://www.gnu.org/licenses/">http://www.gnu.org/licenses/</a>>.</p>
</div></blockquote>
<div class="section" id="abstract">
<h2>Abstract<a class="headerlink" href="#abstract" title="Permalink to this headline">¶</a></h2>
<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>
</div>
<div class="section" id="requirements">
<h2>Requirements<a class="headerlink" href="#requirements" title="Permalink to this headline">¶</a></h2>
<ul class="simple">
<li><a class="reference external" href="http://www.python.org">CPython 2.7 or 3.2</a></li>
<li><a class="reference external" href="http://numpy.scipy.org">Numpy 1.6</a></li>
<li><a class="reference external" href="http://www.lfd.uci.edu/~gohlke/">transformations.c 2012.01.01</a>
(optional implementation of some functions in C)</li>
</ul>
</div>
<div class="section" id="notes">
<h2>Notes<a class="headerlink" href="#notes" title="Permalink to this headline">¶</a></h2>
<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), be concatenated using
numpy.dot(M0, M1), or transform homogeneous coordinate arrays (v) using
numpy.dot(M, v) for shape (4, *) column vectors, respectively
numpy.dot(v, M.T) for shape (*, 4) row vectors (“array of points”).</p>
<p>This module follows the “column vectors on the right” and “row major storage”
(C contiguous) conventions. The translation components are in the right column
of the transformation matrix, i.e. M[:3, 3].
The transpose of the transformation matrices may have to be used to interface
with other graphics systems, e.g. with OpenGL’s glMultMatrixd(). See also [16].</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>first character : rotations are applied to ‘s’tatic or ‘r’otating frame</li>
<li>remaining characters : successive rotation axis ‘x’, ‘y’, or ‘z’</li>
</ul>
<p><em>Axes 4-tuple</em>: e.g. (0, 0, 0, 0) or (1, 1, 1, 1)</p>
<ul class="simple">
<li>inner axis: code of axis (‘x’:0, ‘y’:1, ‘z’:2) of rightmost matrix.</li>
<li>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).</li>
<li>repetition : first and last axis are same (1) or different (0).</li>
<li>frame : rotations are applied to static (0) or rotating (1) frame.</li>
</ul>
</div></blockquote>
</div>
<div class="section" id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
<ol class="arabic simple">
<li>Matrices and transformations. Ronald Goldman.
In “Graphics Gems I”, pp 472-475. Morgan Kaufmann, 1990.</li>
<li>More matrices and transformations: shear and pseudo-perspective.
Ronald Goldman. In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.</li>
<li>Decomposing a matrix into simple transformations. Spencer Thomas.
In “Graphics Gems II”, pp 320-323. Morgan Kaufmann, 1991.</li>
<li>Recovering the data from the transformation matrix. Ronald Goldman.
In “Graphics Gems II”, pp 324-331. Morgan Kaufmann, 1991.</li>
<li>Euler angle conversion. Ken Shoemake.
In “Graphics Gems IV”, pp 222-229. Morgan Kaufmann, 1994.</li>
<li>Arcball rotation control. Ken Shoemake.
In “Graphics Gems IV”, pp 175-192. Morgan Kaufmann, 1994.</li>
<li>Representing attitude: Euler angles, unit quaternions, and rotation
vectors. James Diebel. 2006.</li>
<li>A discussion of the solution for the best rotation to relate two sets
of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828.</li>
<li>Closed-form solution of absolute orientation using unit quaternions.
BKP Horn. J Opt Soc Am A. 1987. 4(4):629-642.</li>
<li>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></li>
<li>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></li>
<li>Uniform random rotations. Ken Shoemake.
In “Graphics Gems III”, pp 124-132. Morgan Kaufmann, 1992.</li>
<li>Quaternion in molecular modeling. CFF Karney.
J Mol Graph Mod, 25(5):595-604</li>
<li>New method for extracting the quaternion from a rotation matrix.
Itzhack Y Bar-Itzhack, J Guid Contr Dynam. 2000. 23(6): 1085-1087.</li>
<li>Multiple View Geometry in Computer Vision. Hartley and Zissermann.
Cambridge University Press; 2nd Ed. 2004. Chapter 4, Algorithm 4.7, p 130.</li>
<li>Column Vectors vs. Row Vectors.
<a class="reference external" href="http://steve.hollasch.net/cgindex/math/matrix/column-vec.html">http://steve.hollasch.net/cgindex/math/matrix/column-vec.html</a></li>
</ol>
</div>
<div class="section" id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
<div class="highlight-default"><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>
<dl class="class">
<dt id="transformations.Arcball">
<em class="property">class </em><code class="descclassname">transformations.</code><code class="descname">Arcball</code><span class="sig-paren">(</span><em>initial=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball" title="Permalink to this definition">¶</a></dt>
<dd><p>Virtual Trackball Control.</p>
<div class="highlight-default"><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="method">
<dt id="transformations.Arcball.down">
<code class="descname">down</code><span class="sig-paren">(</span><em>point</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.down"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.down" title="Permalink to this definition">¶</a></dt>
<dd><p>Set initial cursor window coordinates and pick constrain-axis.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.drag">
<code class="descname">drag</code><span class="sig-paren">(</span><em>point</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.drag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.drag" title="Permalink to this definition">¶</a></dt>
<dd><p>Update current cursor window coordinates.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.getconstrain">
<code class="descname">getconstrain</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.getconstrain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.getconstrain" title="Permalink to this definition">¶</a></dt>
<dd><p>Return state of constrain to axis mode.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.matrix">
<code class="descname">matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return homogeneous rotation matrix.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.next">
<code class="descname">next</code><span class="sig-paren">(</span><em>acceleration=0.0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.next"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.next" title="Permalink to this definition">¶</a></dt>
<dd><p>Continue rotation in direction of last drag.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.place">
<code class="descname">place</code><span class="sig-paren">(</span><em>center</em>, <em>radius</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.place"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.place" title="Permalink to this definition">¶</a></dt>
<dd><p>Place Arcball, e.g. when window size changes.</p>
<dl class="docutils">
<dt>center <span class="classifier-delimiter">:</span> <span class="classifier">sequence[2]</span></dt>
<dd>Window coordinates of trackball center.</dd>
<dt>radius <span class="classifier-delimiter">:</span> <span class="classifier">float</span></dt>
<dd>Radius of trackball in window coordinates.</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.setaxes">
<code class="descname">setaxes</code><span class="sig-paren">(</span><em>*axes</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.setaxes"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.setaxes" title="Permalink to this definition">¶</a></dt>
<dd><p>Set axes to constrain rotations.</p>
</dd></dl>
<dl class="method">
<dt id="transformations.Arcball.setconstrain">
<code class="descname">setconstrain</code><span class="sig-paren">(</span><em>constrain</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#Arcball.setconstrain"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.Arcball.setconstrain" title="Permalink to this definition">¶</a></dt>
<dd><p>Set state of constrain to axis mode.</p>
</dd></dl>
</dd></dl>
<dl class="function">
<dt id="transformations.affine_matrix_from_points">
<code class="descclassname">transformations.</code><code class="descname">affine_matrix_from_points</code><span class="sig-paren">(</span><em>v0</em>, <em>v1</em>, <em>shear=True</em>, <em>scale=True</em>, <em>usesvd=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#affine_matrix_from_points"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.affine_matrix_from_points" title="Permalink to this definition">¶</a></dt>
<dd><p>Return affine transform matrix to register two point sets.</p>
<p>v0 and v1 are shape (ndims, *) arrays of at least ndims non-homogeneous
coordinates, where ndims is the dimensionality of the coordinate space.</p>
<p>If shear is False, a similarity transformation matrix is returned.
If also scale is False, a rigid/Eucledian transformation matrix
is returned.</p>
<p>By default the algorithm by Hartley and Zissermann [15] is used.
If usesvd is True, similarity and Eucledian transformation matrices
are calculated by minimizing the weighted sum of squared deviations
(RMSD) according to the algorithm by Kabsch [8].
Otherwise, and if ndims is 3, the quaternion based algorithm by Horn [9]
is used, which is slower when using this Python implementation.</p>
<p>The returned matrix performs rotation, translation and uniform scaling
(if specified).</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v0</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1031</span><span class="p">,</span> <span class="mi">1031</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">1600</span><span class="p">,</span> <span class="mi">1600</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">675</span><span class="p">,</span> <span class="mi">826</span><span class="p">,</span> <span class="mi">826</span><span class="p">,</span> <span class="mi">677</span><span class="p">],</span> <span class="p">[</span><span class="mi">55</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">281</span><span class="p">,</span> <span class="mi">277</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">affine_matrix_from_points</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="go">array([[ 0.14549, 0.00062, 675.50008],</span>
<span class="go"> [ 0.00048, 0.14094, 53.24971],</span>
<span class="go"> [ 0. , 0. , 1. ]])</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="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="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">random</span><span class="p">(</span><span class="mi">3</span><span class="p">))</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="n">random</span><span class="o">.</span><span class="n">random</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">S</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">100</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="mi">20</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="mi">1</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">M</span><span class="p">,</span> <span class="n">v0</span><span class="p">)</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="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1e-8</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">affine_matrix_from_points</span><span class="p">(</span><span class="n">v0</span><span class="p">[:</span><span class="mi">3</span><span class="p">],</span> <span class="n">v1</span><span class="p">[:</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">v1</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">v0</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
<p>More examples in superimposition_matrix()</p>
</dd></dl>
<dl class="function">
<dt id="transformations.angle_between_vectors">
<code class="descclassname">transformations.</code><code class="descname">angle_between_vectors</code><span class="sig-paren">(</span><em>v0</em>, <em>v1</em>, <em>directed=True</em>, <em>axis=0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#angle_between_vectors"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.angle_between_vectors" title="Permalink 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="highlight-default"><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="mi">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="function">
<dt id="transformations.arcball_constrain_to_axis">
<code class="descclassname">transformations.</code><code class="descname">arcball_constrain_to_axis</code><span class="sig-paren">(</span><em>point</em>, <em>axis</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#arcball_constrain_to_axis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.arcball_constrain_to_axis" title="Permalink to this definition">¶</a></dt>
<dd><p>Return sphere point perpendicular to axis.</p>
</dd></dl>
<dl class="function">
<dt id="transformations.arcball_map_to_sphere">
<code class="descclassname">transformations.</code><code class="descname">arcball_map_to_sphere</code><span class="sig-paren">(</span><em>point</em>, <em>center</em>, <em>radius</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#arcball_map_to_sphere"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.arcball_map_to_sphere" title="Permalink to this definition">¶</a></dt>
<dd><p>Return unit sphere coordinates from window coordinates.</p>
</dd></dl>
<dl class="function">
<dt id="transformations.arcball_nearest_axis">
<code class="descclassname">transformations.</code><code class="descname">arcball_nearest_axis</code><span class="sig-paren">(</span><em>point</em>, <em>axes</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#arcball_nearest_axis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.arcball_nearest_axis" title="Permalink to this definition">¶</a></dt>
<dd><p>Return axis, which arc is nearest to point.</p>
</dd></dl>
<dl class="function">
<dt id="transformations.clip_matrix">
<code class="descclassname">transformations.</code><code class="descname">clip_matrix</code><span class="sig-paren">(</span><em>left</em>, <em>right</em>, <em>bottom</em>, <em>top</em>, <em>near</em>, <em>far</em>, <em>perspective=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#clip_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.clip_matrix" title="Permalink 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 (divided by w coordinate).</p>
<div class="highlight-default"><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="mi">1</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="mi">1</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="mi">1</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="mi">1</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="function">
<dt id="transformations.compose_matrix">
<code class="descclassname">transformations.</code><code class="descname">compose_matrix</code><span class="sig-paren">(</span><em>scale=None</em>, <em>shear=None</em>, <em>angles=None</em>, <em>translate=None</em>, <em>perspective=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#compose_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.compose_matrix" title="Permalink 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="docutils">
<dt>Sequence of transformations:</dt>
<dd>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</dd>
</dl>
<div class="highlight-default"><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="function">
<dt id="transformations.concatenate_matrices">
<code class="descclassname">transformations.</code><code class="descname">concatenate_matrices</code><span class="sig-paren">(</span><em>*matrices</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#concatenate_matrices"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.concatenate_matrices" title="Permalink to this definition">¶</a></dt>
<dd><p>Return concatenation of series of transformation matrices.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.decompose_matrix">
<code class="descclassname">transformations.</code><code class="descname">decompose_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#decompose_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.decompose_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return sequence of transformations from transformation matrix.</p>
<dl class="docutils">
<dt>matrix <span class="classifier-delimiter">:</span> <span class="classifier">array_like</span></dt>
<dd>Non-degenerative homogeneous transformation matrix</dd>
<dt>Return tuple of:</dt>
<dd>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</dd>
</dl>
<p>Raise ValueError if matrix is of wrong type or degenerative.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.euler_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">euler_from_matrix</code><span class="sig-paren">(</span><em>matrix</em>, <em>axes='sxyz'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#euler_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.euler_from_matrix" title="Permalink 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="highlight-default"><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="mi">4</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="function">
<dt id="transformations.euler_from_quaternion">
<code class="descclassname">transformations.</code><code class="descname">euler_from_quaternion</code><span class="sig-paren">(</span><em>quaternion</em>, <em>axes='sxyz'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#euler_from_quaternion"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.euler_from_quaternion" title="Permalink to this definition">¶</a></dt>
<dd><p>Return Euler angles from quaternion for specified axis sequence.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.euler_matrix">
<code class="descclassname">transformations.</code><code class="descname">euler_matrix</code><span class="sig-paren">(</span><em>ai</em>, <em>aj</em>, <em>ak</em>, <em>axes='sxyz'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#euler_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.euler_matrix" title="Permalink 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="highlight-default"><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="mi">4</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="function">
<dt id="transformations.identity_matrix">
<code class="descclassname">transformations.</code><code class="descname">identity_matrix</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#identity_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.identity_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return 4x4 identity/unit matrix.</p>
<div class="highlight-default"><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="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.inverse_matrix">
<code class="descclassname">transformations.</code><code class="descname">inverse_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#inverse_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.inverse_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return inverse of square transformation matrix.</p>
<div class="highlight-default"><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="nb">range</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="function">
<dt id="transformations.is_same_transform">
<code class="descclassname">transformations.</code><code class="descname">is_same_transform</code><span class="sig-paren">(</span><em>matrix0</em>, <em>matrix1</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#is_same_transform"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.is_same_transform" title="Permalink to this definition">¶</a></dt>
<dd><p>Return True if two matrices perform same transformation.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.orthogonalization_matrix">
<code class="descclassname">transformations.</code><code class="descname">orthogonalization_matrix</code><span class="sig-paren">(</span><em>lengths</em>, <em>angles</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#orthogonalization_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.orthogonalization_matrix" title="Permalink 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="highlight-default"><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="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="p">[</span><span class="mi">90</span><span class="p">,</span> <span class="mi">90</span><span class="p">,</span> <span class="mi">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="function">
<dt id="transformations.projection_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">projection_from_matrix</code><span class="sig-paren">(</span><em>matrix</em>, <em>pseudo=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#projection_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.projection_from_matrix" title="Permalink 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="highlight-default"><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="function">
<dt id="transformations.projection_matrix">
<code class="descclassname">transformations.</code><code class="descname">projection_matrix</code><span class="sig-paren">(</span><em>point</em>, <em>normal</em>, <em>direction=None</em>, <em>perspective=None</em>, <em>pseudo=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#projection_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.projection_matrix" title="Permalink 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="highlight-default"><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="mi">20</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="mi">1</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="mi">3</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="function">
<dt id="transformations.quaternion_about_axis">
<code class="descclassname">transformations.</code><code class="descname">quaternion_about_axis</code><span class="sig-paren">(</span><em>angle</em>, <em>axis</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_about_axis"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_about_axis" title="Permalink to this definition">¶</a></dt>
<dd><p>Return quaternion for rotation about axis.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.quaternion_conjugate">
<code class="descclassname">transformations.</code><code class="descname">quaternion_conjugate</code><span class="sig-paren">(</span><em>quaternion</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_conjugate"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_conjugate" title="Permalink to this definition">¶</a></dt>
<dd><p>Return conjugate of quaternion.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.quaternion_from_euler">
<code class="descclassname">transformations.</code><code class="descname">quaternion_from_euler</code><span class="sig-paren">(</span><em>ai</em>, <em>aj</em>, <em>ak</em>, <em>axes='sxyz'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_from_euler"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_from_euler" title="Permalink 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="highlight-default"><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="function">
<dt id="transformations.quaternion_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">quaternion_from_matrix</code><span class="sig-paren">(</span><em>matrix</em>, <em>isprecise=False</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return quaternion from rotation matrix.</p>
<p>If isprecise is True, the input matrix is assumed to be a precise rotation
matrix and a faster algorithm is used.</p>
<div class="highlight-default"><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">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="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="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>
<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="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="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="function">
<dt id="transformations.quaternion_imag">
<code class="descclassname">transformations.</code><code class="descname">quaternion_imag</code><span class="sig-paren">(</span><em>quaternion</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_imag"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_imag" title="Permalink to this definition">¶</a></dt>
<dd><p>Return imaginary part of quaternion.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">quaternion_imag</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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="go">array([ 0., 1., 2.])</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.quaternion_inverse">
<code class="descclassname">transformations.</code><code class="descname">quaternion_inverse</code><span class="sig-paren">(</span><em>quaternion</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_inverse"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_inverse" title="Permalink to this definition">¶</a></dt>
<dd><p>Return inverse of quaternion.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.quaternion_matrix">
<code class="descclassname">transformations.</code><code class="descname">quaternion_matrix</code><span class="sig-paren">(</span><em>quaternion</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return homogeneous rotation matrix from quaternion.</p>
<div class="highlight-default"><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">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">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="function">
<dt id="transformations.quaternion_multiply">
<code class="descclassname">transformations.</code><code class="descname">quaternion_multiply</code><span class="sig-paren">(</span><em>quaternion1</em>, <em>quaternion0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_multiply"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_multiply" title="Permalink to this definition">¶</a></dt>
<dd><p>Return multiplication of two quaternions.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.quaternion_real">
<code class="descclassname">transformations.</code><code class="descname">quaternion_real</code><span class="sig-paren">(</span><em>quaternion</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_real"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_real" title="Permalink to this definition">¶</a></dt>
<dd><p>Return real part of quaternion.</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">quaternion_real</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">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="go">3.0</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.quaternion_slerp">
<code class="descclassname">transformations.</code><code class="descname">quaternion_slerp</code><span class="sig-paren">(</span><em>quat0</em>, <em>quat1</em>, <em>fraction</em>, <em>spin=0</em>, <em>shortestpath=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#quaternion_slerp"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.quaternion_slerp" title="Permalink to this definition">¶</a></dt>
<dd><p>Return spherical linear interpolation between two quaternions.</p>
<div class="highlight-default"><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="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="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="mi">1</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="mi">2</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="mi">2</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="function">
<dt id="transformations.random_quaternion">
<code class="descclassname">transformations.</code><code class="descname">random_quaternion</code><span class="sig-paren">(</span><em>rand=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#random_quaternion"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.random_quaternion" title="Permalink to this definition">¶</a></dt>
<dd><p>Return uniform random unit quaternion.</p>
<dl class="docutils">
<dt>rand: array like or None</dt>
<dd>Three independent random variables that are uniformly distributed
between 0 and 1.</dd>
</dl>
<div class="highlight-default"><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="mi">1</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="function">
<dt id="transformations.random_rotation_matrix">
<code class="descclassname">transformations.</code><code class="descname">random_rotation_matrix</code><span class="sig-paren">(</span><em>rand=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#random_rotation_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.random_rotation_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return uniform random rotation matrix.</p>
<dl class="docutils">
<dt>rand: array like</dt>
<dd>Three independent random variables that are uniformly distributed
between 0 and 1 for each returned quaternion.</dd>
</dl>
<div class="highlight-default"><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="function">
<dt id="transformations.random_vector">
<code class="descclassname">transformations.</code><code class="descname">random_vector</code><span class="sig-paren">(</span><em>size</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#random_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.random_vector" title="Permalink to this definition">¶</a></dt>
<dd><p>Return array of random doubles in the half-open interval [0.0, 1.0).</p>
<div class="highlight-default"><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="mi">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="mi">1</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="function">
<dt id="transformations.reflection_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">reflection_from_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#reflection_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.reflection_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return mirror plane point and normal vector from reflection matrix.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.reflection_matrix">
<code class="descclassname">transformations.</code><code class="descname">reflection_matrix</code><span class="sig-paren">(</span><em>point</em>, <em>normal</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#reflection_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.reflection_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return matrix to mirror at plane defined by point and normal vector.</p>
<div class="highlight-default"><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.</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="mi">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="function">
<dt id="transformations.rotation_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">rotation_from_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#rotation_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.rotation_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return rotation angle and axis from rotation matrix.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.rotation_matrix">
<code class="descclassname">transformations.</code><code class="descname">rotation_matrix</code><span class="sig-paren">(</span><em>angle</em>, <em>direction</em>, <em>point=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#rotation_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.rotation_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return matrix to rotate about axis defined by point and direction.</p>
<div class="highlight-default"><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="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">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="mi">1</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">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="mi">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="function">
<dt id="transformations.scale_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">scale_from_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#scale_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.scale_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return scaling factor, origin and direction from scaling matrix.</p>
<div class="highlight-default"><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="function">
<dt id="transformations.scale_matrix">
<code class="descclassname">transformations.</code><code class="descname">scale_matrix</code><span class="sig-paren">(</span><em>factor</em>, <em>origin=None</em>, <em>direction=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#scale_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.scale_matrix" title="Permalink 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="highlight-default"><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="mi">20</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="mi">1</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>
<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">S</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">S</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>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.shear_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">shear_from_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#shear_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.shear_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return shear angle, direction and plane from shear matrix.</p>
<div class="highlight-default"><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="mi">4</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</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">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">cross</span><span class="p">(</span><span class="n">direct</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="n">S0</span> <span class="o">=</span> <span class="n">shear_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direct</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">angle</span><span class="p">,</span> <span class="n">direct</span><span class="p">,</span> <span class="n">point</span><span class="p">,</span> <span class="n">normal</span> <span class="o">=</span> <span class="n">shear_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">shear_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direct</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">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="function">
<dt id="transformations.shear_matrix">
<code class="descclassname">transformations.</code><code class="descname">shear_matrix</code><span class="sig-paren">(</span><em>angle</em>, <em>direction</em>, <em>point</em>, <em>normal</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#shear_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.shear_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return matrix to shear by angle along direction vector on shear plane.</p>
<p>The shear plane is defined by a point and normal vector. The direction
vector must be orthogonal to the plane’s normal vector.</p>
<p>A point P is transformed by the shear matrix into P” such that
the vector P-P” is parallel to the direction vector and its extent is
given by the angle of P-P’-P”, where P’ is the orthogonal projection
of P onto the shear plane.</p>
<div class="highlight-default"><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="mi">4</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">pi</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">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">cross</span><span class="p">(</span><span class="n">direct</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="n">S</span> <span class="o">=</span> <span class="n">shear_matrix</span><span class="p">(</span><span class="n">angle</span><span class="p">,</span> <span class="n">direct</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">numpy</span><span class="o">.</span><span class="n">allclose</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">linalg</span><span class="o">.</span><span class="n">det</span><span class="p">(</span><span class="n">S</span><span class="p">))</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.superimposition_matrix">
<code class="descclassname">transformations.</code><code class="descname">superimposition_matrix</code><span class="sig-paren">(</span><em>v0</em>, <em>v1</em>, <em>scale=False</em>, <em>usesvd=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#superimposition_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.superimposition_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return matrix to transform given 3D point set into second point set.</p>
<p>v0 and v1 are shape (3, *) or (4, *) arrays of at least 3 points.</p>
<p>The parameters scale and usesvd are explained in the more general
affine_matrix_from_points function.</p>
<p>The returned matrix is a similarity or Eucledian transformation matrix.
This function has a fast C implementation in transformations.c.</p>
<div class="highlight-default"><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">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">superimposition_matrix</span><span class="p">(</span><span class="n">v0</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">M</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">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">random</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v0</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="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="p">[</span><span class="mi">1</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="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">R</span><span class="p">,</span> <span class="n">v0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">superimposition_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="n">v1</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">v0</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="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">100</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="mi">20</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="mi">1</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">R</span><span class="p">,</span> <span class="n">v0</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">superimposition_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="n">v1</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">v0</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="n">random</span><span class="o">.</span><span class="n">random</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="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="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">S</span><span class="p">)</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">M</span><span class="p">,</span> <span class="n">v0</span><span class="p">)</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="n">numpy</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">1e-9</span><span class="p">,</span> <span class="mi">300</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">superimposition_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="n">scale</span><span class="o">=</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">v1</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">v0</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">superimposition_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="n">scale</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">usesvd</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">v1</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">v0</span><span class="p">))</span>
<span class="go">True</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">empty</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">v</span><span class="p">[:,</span> <span class="p">:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">v0</span>
<span class="gp">>>> </span><span class="n">M</span> <span class="o">=</span> <span class="n">superimposition_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="n">scale</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">usesvd</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">v1</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">v</span><span class="p">[:,</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="function">
<dt id="transformations.translation_from_matrix">
<code class="descclassname">transformations.</code><code class="descname">translation_from_matrix</code><span class="sig-paren">(</span><em>matrix</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#translation_from_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.translation_from_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return translation vector from translation matrix.</p>
<div class="highlight-default"><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">translation_from_matrix</span><span class="p">(</span><span class="n">translation_matrix</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">v0</span><span class="p">,</span> <span class="n">v1</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.translation_matrix">
<code class="descclassname">transformations.</code><code class="descname">translation_matrix</code><span class="sig-paren">(</span><em>direction</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#translation_matrix"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.translation_matrix" title="Permalink to this definition">¶</a></dt>
<dd><p>Return matrix to translate by direction vector.</p>
<div class="highlight-default"><div class="highlight"><pre><span></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">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">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">translation_matrix</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="mi">3</span><span class="p">])</span>
<span class="go">True</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.unit_vector">
<code class="descclassname">transformations.</code><code class="descname">unit_vector</code><span class="sig-paren">(</span><em>data</em>, <em>axis=None</em>, <em>out=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#unit_vector"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.unit_vector" title="Permalink to this definition">¶</a></dt>
<dd><p>Return ndarray normalized by length, i.e. eucledian norm, along axis.</p>
<div class="highlight-default"><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="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">unit_vector</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="n">v0</span> <span class="o">/</span> <span class="n">numpy</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">norm</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">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">rand</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">unit_vector</span><span class="p">(</span><span class="n">v0</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">v2</span> <span class="o">=</span> <span class="n">v0</span> <span class="o">/</span> <span class="n">numpy</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sqrt</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">v0</span><span class="o">*</span><span class="n">v0</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</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">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="n">v2</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">v1</span> <span class="o">=</span> <span class="n">unit_vector</span><span class="p">(</span><span class="n">v0</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">v2</span> <span class="o">=</span> <span class="n">v0</span> <span class="o">/</span> <span class="n">numpy</span><span class="o">.</span><span class="n">expand_dims</span><span class="p">(</span><span class="n">numpy</span><span class="o">.</span><span class="n">sqrt</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">v0</span><span class="o">*</span><span class="n">v0</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="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">v1</span><span class="p">,</span> <span class="n">v2</span><span class="p">)</span>
<span class="go">True</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">empty</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">unit_vector</span><span class="p">(</span><span class="n">v0</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="n">out</span><span class="o">=</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">v1</span><span class="p">,</span> <span class="n">v2</span><span class="p">)</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="n">unit_vector</span><span class="p">([]))</span>
<span class="go">[]</span>
<span class="gp">>>> </span><span class="nb">list</span><span class="p">(</span><span class="n">unit_vector</span><span class="p">([</span><span class="mi">1</span><span class="p">]))</span>
<span class="go">[1.0]</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.vector_norm">
<code class="descclassname">transformations.</code><code class="descname">vector_norm</code><span class="sig-paren">(</span><em>data</em>, <em>axis=None</em>, <em>out=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#vector_norm"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.vector_norm" title="Permalink to this definition">¶</a></dt>
<dd><p>Return length, i.e. eucledian norm, of ndarray along axis.</p>
<div class="highlight-default"><div class="highlight"><pre><span></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">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="n">n</span> <span class="o">=</span> <span class="n">vector_norm</span><span class="p">(</span><span class="n">v</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">n</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">norm</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
<span class="go">True</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">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="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="n">vector_norm</span><span class="p">(</span><span class="n">v</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">n</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sqrt</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">v</span><span class="o">*</span><span class="n">v</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">2</span><span class="p">)))</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="n">vector_norm</span><span class="p">(</span><span class="n">v</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">n</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sqrt</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">v</span><span class="o">*</span><span class="n">v</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="go">True</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">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">vector_norm</span><span class="p">(</span><span class="n">v</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="n">out</span><span class="o">=</span><span class="n">n</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">n</span><span class="p">,</span> <span class="n">numpy</span><span class="o">.</span><span class="n">sqrt</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">v</span><span class="o">*</span><span class="n">v</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="go">True</span>
<span class="gp">>>> </span><span class="n">vector_norm</span><span class="p">([])</span>
<span class="go">0.0</span>
<span class="gp">>>> </span><span class="n">vector_norm</span><span class="p">([</span><span class="mi">1</span><span class="p">])</span>
<span class="go">1.0</span>
</pre></div>
</div>
</dd></dl>
<dl class="function">
<dt id="transformations.vector_product">
<code class="descclassname">transformations.</code><code class="descname">vector_product</code><span class="sig-paren">(</span><em>v0</em>, <em>v1</em>, <em>axis=0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/transformations.html#vector_product"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#transformations.vector_product" title="Permalink to this definition">¶</a></dt>
<dd><p>Return vector perpendicular to vectors (cross product).</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="n">vector_product</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">3</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">v</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">6</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">v</span> <span class="o">=</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">numpy</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">v</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">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">6</span><span class="p">,</span> <span class="mi">6</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">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="mi">6</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">v</span> <span class="o">=</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="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">v</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">6</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">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">6</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="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">]])</span>
<span class="go">True</span>
</pre></div>
</div>
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