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<div class="section" id="glossary">
<h1>Glossary<a class="headerlink" href="#glossary" title="Permalink to this headline">¶</a></h1>
<dl class="glossary docutils">
<dt id="term-affine-matrix">Affine matrix</dt>
<dd>A matrix implementing an <a class="reference internal" href="#term-affine-transformation"><span class="xref std std-term">affine transformation</span></a> in
<a class="reference internal" href="#term-homogenous-coordinates"><span class="xref std std-term">homogenous coordinates</span></a>. For a 3 dimensional transform, the
matrix is shape 4 by 4.</dd>
<dt id="term-affine-transformation">Affine transformation</dt>
<dd>See <a class="reference external" href="http://en.wikipedia.org/wiki/Affine_transformation">wikipedia affine</a> definition. An affine transformation is a
<a class="reference internal" href="#term-linear-transformation"><span class="xref std std-term">linear transformation</span></a> followed by a translation.</dd>
<dt id="term-axis-angle">Axis angle</dt>
<dd>A representation of rotation. See: <a class="reference external" href="http://en.wikipedia.org/wiki/Axis_angle">wikipedia axis angle</a> .
From Euler’s rotation theorem we know that any rotation or
sequence of rotations can be represented by a single rotation
about an axis. The axis <span class="math">\(\boldsymbol{\hat{u}}\)</span> is a <a class="reference internal" href="#term-unit-vector"><span class="xref std std-term">unit
vector</span></a>. The angle is <span class="math">\(\theta\)</span>. The <a class="reference internal" href="#term-rotation-vector"><span class="xref std std-term">rotation vector</span></a> is a
more compact representation of <span class="math">\(\theta\)</span> and
<span class="math">\(\boldsymbol{\hat{u}}\)</span>.</dd>
<dt id="term-euclidean-norm">Euclidean norm</dt>
<dd><p class="first">Also called Euclidean length, or L2 norm. The Euclidean norm
<span class="math">\(\|\mathbf{x}\|\)</span> of a vector <span class="math">\(\mathbf{x}\)</span> is given by:</p>
<div class="math">
\[\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}\]</div>
<p class="last">Pure Pythagoras.</p>
</dd>
<dt id="term-euler-angles">Euler angles</dt>
<dd>See: <a class="reference external" href="http://en.wikipedia.org/wiki/Euler_angles">wikipedia Euler angles</a> and <a class="reference external" href="http://mathworld.wolfram.com/EulerAngles.html">Mathworld Euler angles</a>.</dd>
<dt id="term-gimbal-lock">Gimbal lock</dt>
<dd>See <a class="reference internal" href="gimbal_lock.html#gimbal-lock"><span class="std std-ref">Gimbal lock</span></a></dd>
<dt id="term-homogenous-coordinates">Homogenous coordinates</dt>
<dd>See <a class="reference external" href="http://en.wikipedia.org/wiki/Homogeneous_coordinates">wikipedia homogenous coordinates</a></dd>
<dt id="term-linear-transformation">Linear transformation</dt>
<dd>A linear transformation is one that preserves lines - that is, if
any three points are on a line before transformation, they are
also on a line after transformation. See <a class="reference external" href="http://en.wikipedia.org/wiki/Linear_transformation">wikipedia linear
transform</a>. Rotation, scaling and shear are linear
transformations.</dd>
<dt id="term-quaternion">Quaternion</dt>
<dd>See: <a class="reference external" href="http://en.wikipedia.org/wiki/Quaternion">wikipedia quaternion</a>. An extension of the complex numbers
that can represent a rotation. Quaternions have 4 values, <span class="math">\(w, x,
y, z\)</span>. <span class="math">\(w\)</span> is the <em>real</em> part of the quaternion and the vector
<span class="math">\(x, y, z\)</span> is the <em>vector</em> part of the quaternion. Quaternions are
less intuitive to visualize than <a class="reference internal" href="#term-euler-angles"><span class="xref std std-term">Euler angles</span></a> but do not
suffer from <a class="reference internal" href="#term-gimbal-lock"><span class="xref std std-term">gimbal lock</span></a> and are often used for rapid
interpolation of rotations.</dd>
<dt id="term-reflection">Reflection</dt>
<dd>A transformation that can be thought of as transforming an object
to its mirror image. The mirror in the transformation is a plane.
A plan can be defined with a point and a vector normal to the
plane. See <a class="reference external" href="http://en.wikipedia.org/wiki/Reflection_(mathematics)">wikipedia reflection</a>.</dd>
<dt id="term-rotation-matrix">Rotation matrix</dt>
<dd>See <a class="reference external" href="http://en.wikipedia.org/wiki/Rotation_matrix">wikipedia rotation matrix</a>. A rotation matrix is a matrix
implementing a rotation. Rotation matrices are square and
orthogonal. That means, that the rotation matrix <span class="math">\(R\)</span> has columns
and rows that are <a class="reference internal" href="#term-unit-vector"><span class="xref std std-term">unit vector</span></a>, and where <span class="math">\(R^T R = I\)</span> (<span class="math">\(R^T\)</span> is
the transpose and <span class="math">\(I\)</span> is the identity matrix). Therefore <span class="math">\(R^T =
R^{-1}\)</span> (<span class="math">\(R^{-1}\)</span> is the inverse). Rotation matrices also have a
determinant of <span class="math">\(1\)</span>.</dd>
<dt id="term-rotation-vector">Rotation vector</dt>
<dd><p class="first">A representation of an <a class="reference internal" href="#term-axis-angle"><span class="xref std std-term">axis angle</span></a> rotation. The angle
<span class="math">\(\theta\)</span> and unit vector axis <span class="math">\(\boldsymbol{\hat{u}}\)</span> are stored in a
<em>rotation vector</em> <span class="math">\(\boldsymbol{u}\)</span>, such that:</p>
<div class="math">
\[ \begin{align}\begin{aligned}\theta = \|\boldsymbol{u}\| \,\\\boldsymbol{\hat{u}} = \frac{\boldsymbol{u}}{\|\boldsymbol{u}\|}\end{aligned}\end{align} \]</div>
<p class="last">where <span class="math">\(\|\boldsymbol{u}\|\)</span> is the <a class="reference internal" href="#term-euclidean-norm"><span class="xref std std-term">Euclidean norm</span></a> of
<span class="math">\(\boldsymbol{u}\)</span></p>
</dd>
<dt id="term-shear-matrix">Shear matrix</dt>
<dd>Square matrix that results in shearing transforms - see
<a class="reference external" href="http://en.wikipedia.org/wiki/Shear_matrix">wikipedia shear matrix</a>.</dd>
<dt id="term-unit-vector">Unit vector</dt>
<dd>A vector <span class="math">\(\boldsymbol{\hat{u}}\)</span> with a <a class="reference internal" href="#term-euclidean-norm"><span class="xref std std-term">Euclidean norm</span></a>
of 1. Normalized vector is a synonym. The “hat” over the
<span class="math">\(\boldsymbol{\hat{u}}\)</span> is a convention to express the fact that it
is a unit vector.</dd>
</dl>
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