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<h1>Quaternion Class Reference</h1><!-- doxytag: class="qglviewer::Quaternion" -->
<p>The Quaternion class represents 3D rotations and orientations.
<a href="#_details">More...</a></p>
<p><a href="classqglviewer_1_1Quaternion-members.html">List of all members.</a></p>
<table border="0" cellpadding="0" cellspacing="0">
<tr><td colspan="2"><h2>Defining a Quaternion</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a65ed15cc19af958b5933b5c522f10e66">Quaternion</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ad0347716e801bcdeab909493817edc85">Quaternion</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &axis, double angle)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a1b60be34a715145efc3b91e6dfba1634">Quaternion</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &from, const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &to)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a81ba24ffd95778f3ca4e51b2b9922f45">Quaternion</a> (double q0, double q1, double q2, double q3)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a71a4d1a3b760854468ff270a982e5f59">Quaternion</a> (const Quaternion &Q)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion & </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ad20a1310f5fac88d5e00fa055e09fe72">operator=</a> (const Quaternion &Q)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ad6bd25e80a18f6bd38ae5c4e5337b53b">setAxisAngle</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &axis, double angle)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#abe7dd94e4fb48e0c8e0d994cac84064a">setValue</a> (double q0, double q1, double q2, double q3)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a1880f20e8080ee670b6a88ed0c7d69ac">setFromRotationMatrix</a> (const double m[3][3])</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a760617f082c950633b5642d2f60b2dd9">setFromRotatedBasis</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &X, const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &Y, const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &Z)</td></tr>
<tr><td colspan="2"><h2>Accessing values</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">float </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a46b7892fe36e140862b9c71e5cfe8292">getAxisAngle</a> (<a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &axis, float &angle) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a605b34d1fc87b2ccaf49a45419c2e46f">operator[]</a> (int i) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">double & </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#af1a8c9c4769161714a5a1a2c7fad446b">operator[]</a> (int i)</td></tr>
<tr><td colspan="2"><h2>Rotation computations</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion & </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a0caef8e6e702101a45fdc0af3920c49d">operator*=</a> (const Quaternion &q)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a5e2421069d88e576334e10f634625815">rotate</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &v) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a74d79bc1afcc02fdbfe0eb390c3546f5">inverseRotate</a> (const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &v) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a76b3ffdb188246ff6559069cb3f5b919">operator*</a> (const Quaternion &a, const Quaternion &b)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a19ce6efe5ef2744c8293e8ba1a39b9e4">operator*</a> (const Quaternion &q, const <a class="el" href="classqglviewer_1_1Vec.html">Vec</a> &v)</td></tr>
<tr><td colspan="2"><h2>Inversion</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a7fa1616cc61c19a5efcc863c950f7f30">invert</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#abcdb1512395327f8236a4f4a4d4ff648">negate</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">double </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a05826e509c686f39baaec4656f1a7231">normalize</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ae347ecf6e0a78618e8fd1a3b9107df32">normalized</a> () const </td></tr>
<tr><td colspan="2"><h2>Associated matrix</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">const GLdouble * </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#aa12123cf45f68b9f7fe0526b70ab0047">matrix</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix</a> (GLdouble m[4][4]) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a15f0dc7596dc78437154302466ac3c0a">getMatrix</a> (GLdouble m[16]) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#aad37aa66da74be6fb8d9d66f059f0872">getRotationMatrix</a> (float m[3][3]) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">const GLdouble * </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a32992f152b7377da431c69f15384ed22">inverseMatrix</a> () const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix</a> (GLdouble m[4][4]) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a315cf039df621a101a953df20acc7155">getInverseMatrix</a> (GLdouble m[16]) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#ae1455acb1cbf59befe65295734d48c4a">getInverseRotationMatrix</a> (float m[3][3]) const </td></tr>
<tr><td colspan="2"><h2>Slerp interpolation</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a4db67fbb8171a5e781b56404112cf848">log</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a5c546a33cc0c65f24b7e9c48e5069ba7">exp</a> ()</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a755201c9a752fd892a3656a517b190e4">slerp</a> (const Quaternion &a, const Quaternion &b, float t, bool allowFlip=true)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a4b218b205d88a4dd50c8de38522b5812">squad</a> (const Quaternion &a, const Quaternion &tgA, const Quaternion &tgB, const Quaternion &b, float t)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static double </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a80d06247e39abf2980e56d2fe8c4bb83">dot</a> (const Quaternion &a, const Quaternion &b)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#af4c74176967acca6e3947977351e1c68">lnDif</a> (const Quaternion &a, const Quaternion &b)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a2a27d43a68da6c7c5243fe81e5a9d005">squadTangent</a> (const Quaternion &before, const Quaternion &center, const Quaternion &after)</td></tr>
<tr><td colspan="2"><h2>XML representation</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a87faf5efc96c9b5af85a611985b6618f">Quaternion</a> (const QDomElement &element)</td></tr>
<tr><td class="memItemLeft" align="right" valign="top">QDomElement </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a48e0e2dd26cd96418c8b889ceabe80f6">domElement</a> (const QString &name, QDomDocument &document) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#acd13d2ddeca530cb9f26ead47f7d25d3">initFromDOMElement</a> (const QDomElement &element)</td></tr>
<tr><td colspan="2"><h2>Output stream</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">std::ostream & </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a16d5f2f021103b05006b0c58fbd48796">operator<<</a> (std::ostream &o, const <a class="el" href="classqglviewer_1_1Vec.html">qglviewer::Vec</a> &)</td></tr>
<tr><td colspan="2"><h2>Random Quaternion</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top">static Quaternion </td><td class="memItemRight" valign="bottom"><a class="el" href="classqglviewer_1_1Quaternion.html#a87d5e98d958fdc63100979fcff1d9976">randomQuaternion</a> ()</td></tr>
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<hr/><a name="_details"></a><h2>Detailed Description</h2>
<p>The Quaternion class represents 3D rotations and orientations. </p>
<p>The Quaternion is an appropriate (although not very intuitive) representation for 3D rotations and orientations. Many tools are provided to ease the definition of a Quaternion: see constructors, <a class="el" href="classqglviewer_1_1Quaternion.html#ad6bd25e80a18f6bd38ae5c4e5337b53b">setAxisAngle()</a>, <a class="el" href="classqglviewer_1_1Quaternion.html#a1880f20e8080ee670b6a88ed0c7d69ac">setFromRotationMatrix()</a>, <a class="el" href="classqglviewer_1_1Quaternion.html#a760617f082c950633b5642d2f60b2dd9">setFromRotatedBasis()</a>.</p>
<p>You can apply the rotation represented by the Quaternion to 3D points using <a class="el" href="classqglviewer_1_1Quaternion.html#a5e2421069d88e576334e10f634625815">rotate()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a74d79bc1afcc02fdbfe0eb390c3546f5">inverseRotate()</a>. See also the <a class="el" href="classqglviewer_1_1Frame.html" title="The Frame class represents a coordinate system, defined by a position and an orientation...">Frame</a> class that represents a coordinate system and provides other conversion functions like <a class="el" href="classqglviewer_1_1Frame.html#ac706b338b08d9d7b511fd84d0b6ecf96">Frame::coordinatesOf()</a> and <a class="el" href="classqglviewer_1_1Frame.html#a05a3e18419f02427366a95b1e299f12e">Frame::transformOf()</a>.</p>
<p>You can apply the Quaternion <code>q</code> rotation to the OpenGL matrices using: </p>
<div class="fragment"><pre class="fragment"> glMultMatrixd(q.matrix());
<span class="comment">// equvalent to glRotate(q.angle()*180.0/M_PI, q.axis().x, q.axis().y, q.axis().z);</span>
</pre></div><p>Quaternion is part of the <code>qglviewer</code> namespace, specify <code>qglviewer::Quaternion</code> or use the qglviewer namespace: </p>
<div class="fragment"><pre class="fragment"> <span class="keyword">using namespace </span>qglviewer;
</pre></div><h3>Internal representation</h3>
<p>The internal representation of a Quaternion corresponding to a rotation around axis <code>axis</code>, with an angle <code>alpha</code> is made of four doubles q[i]: </p>
<div class="fragment"><pre class="fragment"> {q[0],q[1],q[2]} = sin(alpha/2) * {<a class="code" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis</a>[0],<a class="code" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis</a>[1],axis[2]}
q[3] = cos(alpha/2)
</pre></div><p>Note that certain implementations place the cosine term in first position (instead of last here).</p>
<p>The Quaternion is always normalized, so that its <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a> is actually its conjugate.</p>
<p>See also the <a class="el" href="classqglviewer_1_1Vec.html" title="The Vec class represents 3D positions and 3D vectors.">Vec</a> and <a class="el" href="classqglviewer_1_1Frame.html" title="The Frame class represents a coordinate system, defined by a position and an orientation...">Frame</a> classes' documentations. </p>
<hr/><h2>Constructor & Destructor Documentation</h2>
<a class="anchor" id="a65ed15cc19af958b5933b5c522f10e66"></a><!-- doxytag: member="qglviewer::Quaternion::Quaternion" ref="a65ed15cc19af958b5933b5c522f10e66" args="()" -->
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<p>Default constructor, builds an identity rotation. </p>
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<a class="anchor" id="ad0347716e801bcdeab909493817edc85"></a><!-- doxytag: member="qglviewer::Quaternion::Quaternion" ref="ad0347716e801bcdeab909493817edc85" args="(const Vec &axis, double angle)" -->
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<p>Constructor from rotation axis (non null) and angle (in radians). See also <a class="el" href="classqglviewer_1_1Quaternion.html#ad6bd25e80a18f6bd38ae5c4e5337b53b">setAxisAngle()</a>. </p>
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<p>Constructs a Quaternion that will rotate from the <code>from</code> direction to the <code>to</code> direction.</p>
<p>Note that this rotation is not uniquely defined. The selected axis is usually orthogonal to <code>from</code> and <code>to</code>, minimizing the rotation angle. This method is robust and can handle small or almost identical vectors. </p>
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<td class="memname">Quaternion </td>
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<td class="paramtype">double </td>
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<td class="paramtype">double </td>
<td class="paramname"> <em>q2</em>, </td>
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<td class="paramtype">double </td>
<td class="paramname"> <em>q3</em></td><td> </td>
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<p>Constructor from the four values of a Quaternion. First three values are axis*sin(angle/2) and last one is cos(angle/2).</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd>The identity Quaternion is Quaternion(0,0,0,1) and <em>not</em> Quaternion(0,0,0,0) (which is not unitary). The default <a class="el" href="classqglviewer_1_1Quaternion.html#a65ed15cc19af958b5933b5c522f10e66">Quaternion()</a> creates such identity Quaternion. </dd></dl>
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<a class="anchor" id="a71a4d1a3b760854468ff270a982e5f59"></a><!-- doxytag: member="qglviewer::Quaternion::Quaternion" ref="a71a4d1a3b760854468ff270a982e5f59" args="(const Quaternion &Q)" -->
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<td class="memname">Quaternion </td>
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<p>Copy constructor. </p>
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<td><code> [explicit]</code></td>
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<p>Constructs a Quaternion from a <code>QDomElement</code> representing an XML code of the form </p>
<div class="fragment"><pre class="fragment"> < anyTagName q0=<span class="stringliteral">".."</span> q1=<span class="stringliteral">".."</span> q2=<span class="stringliteral">".."</span> q3=<span class="stringliteral">".."</span> />
</pre></div><p>If one of these attributes is missing or is not a number, a warning is displayed and the associated value is respectively set to 0, 0, 0 and 1 (identity Quaternion).</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a48e0e2dd26cd96418c8b889ceabe80f6">domElement()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#acd13d2ddeca530cb9f26ead47f7d25d3">initFromDOMElement()</a>. </p>
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<hr/><h2>Member Function Documentation</h2>
<a class="anchor" id="a39bde10d142ce0cbc9614988ca0a0e59"></a><!-- doxytag: member="qglviewer::Quaternion::angle" ref="a39bde10d142ce0cbc9614988ca0a0e59" args="() const " -->
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<p>Returns the angle (in radians) of the rotation represented by the Quaternion.</p>
<p>This value is always in the range [0-pi]. Larger rotational angles are obtained by inverting the <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a> direction.</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a46b7892fe36e140862b9c71e5cfe8292">getAxisAngle()</a>. </p>
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<td class="memname"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> axis </td>
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<p>Returns the normalized axis direction of the rotation represented by the Quaternion.</p>
<p>It is null for an identity Quaternion. See also <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a46b7892fe36e140862b9c71e5cfe8292">getAxisAngle()</a>. </p>
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<p>Returns an XML <code>QDomElement</code> that represents the Quaternion.</p>
<p><code>name</code> is the name of the QDomElement tag. <code>doc</code> is the <code>QDomDocument</code> factory used to create QDomElement.</p>
<p>When output to a file, the resulting QDomElement will look like: </p>
<div class="fragment"><pre class="fragment"> <name q0=<span class="stringliteral">".."</span> q1=<span class="stringliteral">".."</span> q2=<span class="stringliteral">".."</span> q3=<span class="stringliteral">".."</span> />
</pre></div><p>Use <a class="el" href="classqglviewer_1_1Quaternion.html#acd13d2ddeca530cb9f26ead47f7d25d3">initFromDOMElement()</a> to restore the Quaternion state from the resulting <code>QDomElement</code>. See also the <a class="el" href="classqglviewer_1_1Quaternion.html#a87faf5efc96c9b5af85a611985b6618f">Quaternion(const QDomElement&)</a> constructor.</p>
<p>See the <a class="el" href="classqglviewer_1_1Vec.html#a48e0e2dd26cd96418c8b889ceabe80f6">Vec::domElement()</a> documentation for a complete QDomDocument creation and saving example.</p>
<p>See also <a class="el" href="classqglviewer_1_1Frame.html#a48e0e2dd26cd96418c8b889ceabe80f6">Frame::domElement()</a>, <a class="el" href="classqglviewer_1_1Camera.html#a48e0e2dd26cd96418c8b889ceabe80f6">Camera::domElement()</a>, <a class="el" href="classqglviewer_1_1KeyFrameInterpolator.html#a48e0e2dd26cd96418c8b889ceabe80f6">KeyFrameInterpolator::domElement()</a>... </p>
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<td class="paramtype">const Quaternion & </td>
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<p>Returns the "dot" product of <code>a</code> and <code>b:</code> a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3]. </p>
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<p>Returns the exponential of the Quaternion. See also <a class="el" href="classqglviewer_1_1Quaternion.html#a4db67fbb8171a5e781b56404112cf848">log()</a>. </p>
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<td class="memname">void getAxisAngle </td>
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<p>Returns the axis vector and the angle (in radians) of the rotation represented by the Quaternion. See the <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a> documentations. </p>
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<a class="anchor" id="a315cf039df621a101a953df20acc7155"></a><!-- doxytag: member="qglviewer::Quaternion::getInverseMatrix" ref="a315cf039df621a101a953df20acc7155" args="(GLdouble m[16]) const " -->
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<p>Same as <a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix()</a>, but with a <code>GLdouble</code>[16] parameter. See also <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a>. </p>
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<p>Fills <code>m</code> with the OpenGL matrix corresponding to the <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a> rotation.</p>
<p>Use <a class="el" href="classqglviewer_1_1Quaternion.html#a32992f152b7377da431c69f15384ed22">inverseMatrix()</a> if you do not need to store this matrix and simply want to alter the current OpenGL matrix. See also <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a>. </p>
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<p><code>m</code> is set to the 3x3 <em>inverse</em> rotation matrix associated with the Quaternion.</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd>This is the classical mathematical rotation matrix. The OpenGL format uses its transposed version. See <a class="el" href="classqglviewer_1_1Quaternion.html#a32992f152b7377da431c69f15384ed22">inverseMatrix()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix()</a>. </dd></dl>
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<p>Same as <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a>, but with a <code>GLdouble</code>[16] parameter. See also <a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix()</a> and <a class="el" href="classqglviewer_1_1Frame.html#a4915640a54a6b48a99cbdd0cd42fec48">Frame::getMatrix()</a>. </p>
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<p>Fills <code>m</code> with the OpenGL representation of the Quaternion rotation.</p>
<p>Use <a class="el" href="classqglviewer_1_1Quaternion.html#aa12123cf45f68b9f7fe0526b70ab0047">matrix()</a> if you do not need to store this matrix and simply want to alter the current OpenGL matrix. See also <a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix()</a> and <a class="el" href="classqglviewer_1_1Frame.html#a4915640a54a6b48a99cbdd0cd42fec48">Frame::getMatrix()</a>. </p>
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<td class="paramtype">float </td>
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<td> const</td>
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<p>Fills <code>m</code> with the 3x3 rotation matrix associated with the Quaternion.</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#ae1455acb1cbf59befe65295734d48c4a">getInverseRotationMatrix()</a>.</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd><code>m</code> uses the European mathematical representation of the rotation matrix. Use <a class="el" href="classqglviewer_1_1Quaternion.html#aa12123cf45f68b9f7fe0526b70ab0047">matrix()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a> to retrieve the OpenGL transposed version. </dd></dl>
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<p>Restores the Quaternion state from a <code>QDomElement</code> created by <a class="el" href="classqglviewer_1_1Quaternion.html#a48e0e2dd26cd96418c8b889ceabe80f6">domElement()</a>.</p>
<p>The <code>QDomElement</code> should contain the <code>q0</code>, <code>q1</code> , <code>q2</code> and <code>q3</code> attributes. If one of these attributes is missing or is not a number, a warning is displayed and these fields are respectively set to 0.0, 0.0, 0.0 and 1.0 (identity Quaternion).</p>
<p>See also the <a class="el" href="classqglviewer_1_1Quaternion.html#a87faf5efc96c9b5af85a611985b6618f">Quaternion(const QDomElement&)</a> constructor. </p>
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<p>Returns the inverse Quaternion (inverse rotation).</p>
<p>Result has a negated <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a> direction and the same <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a>. A composition (see <a class="el" href="classqglviewer_1_1Quaternion.html#a76b3ffdb188246ff6559069cb3f5b919">operator*()</a>) of a Quaternion and its <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a> results in an identity function.</p>
<p>Use <a class="el" href="classqglviewer_1_1Quaternion.html#a7fa1616cc61c19a5efcc863c950f7f30">invert()</a> to actually modify the Quaternion. </p>
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<p>Returns the associated 4x4 OpenGL <em>inverse</em> rotation matrix. This is simply the <a class="el" href="classqglviewer_1_1Quaternion.html#aa12123cf45f68b9f7fe0526b70ab0047">matrix()</a> of the <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a>.</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd>The result is only valid until the next call to <a class="el" href="classqglviewer_1_1Quaternion.html#a32992f152b7377da431c69f15384ed22">inverseMatrix()</a>. Use it immediately (as in <code>glMultMatrixd</code>(q.inverseMatrix())) or use <a class="el" href="classqglviewer_1_1Quaternion.html#aee97ea226e31ac805a2245e7c73dac7b">getInverseMatrix()</a> instead.</dd>
<dd>
The matrix is given in OpenGL format (row-major order) and is the transpose of the actual mathematical European representation. Consider using <a class="el" href="classqglviewer_1_1Quaternion.html#ae1455acb1cbf59befe65295734d48c4a">getInverseRotationMatrix()</a> instead. </dd></dl>
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<td class="memname"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> inverseRotate </td>
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<p>Returns the image of <code>v</code> by the Quaternion <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a> rotation.</p>
<p><a class="el" href="classqglviewer_1_1Quaternion.html#a5e2421069d88e576334e10f634625815">rotate()</a> performs an inverse transformation. Same as <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a>.rotate(v). </p>
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<p>Inverses the Quaternion (same rotation <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a>, but negated <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a>).</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a>. </p>
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<p>Returns log(a. <a class="el" href="classqglviewer_1_1Quaternion.html#ac378ebc684d691dc25336b895f24a82e">inverse()</a> * b). Useful for <a class="el" href="classqglviewer_1_1Quaternion.html#a2a27d43a68da6c7c5243fe81e5a9d005">squadTangent()</a>. </p>
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<p>Returns the logarithm of the Quaternion. See also <a class="el" href="classqglviewer_1_1Quaternion.html#a5c546a33cc0c65f24b7e9c48e5069ba7">exp()</a>. </p>
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<p>Returns the Quaternion associated 4x4 OpenGL rotation matrix.</p>
<p>Use <code>glMultMatrixd</code>(q.matrix()) to apply the rotation represented by Quaternion <code>q</code> to the current OpenGL matrix.</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a>, <a class="el" href="classqglviewer_1_1Quaternion.html#aad37aa66da74be6fb8d9d66f059f0872">getRotationMatrix()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a32992f152b7377da431c69f15384ed22">inverseMatrix()</a>.</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd>The result is only valid until the next call to <a class="el" href="classqglviewer_1_1Quaternion.html#aa12123cf45f68b9f7fe0526b70ab0047">matrix()</a>. Use it immediately (as shown above) or consider using <a class="el" href="classqglviewer_1_1Quaternion.html#a4915640a54a6b48a99cbdd0cd42fec48">getMatrix()</a> instead.</dd>
<dd>
The matrix is given in OpenGL format (row-major order) and is the transpose of the actual mathematical European representation. Consider using <a class="el" href="classqglviewer_1_1Quaternion.html#aad37aa66da74be6fb8d9d66f059f0872">getRotationMatrix()</a> instead. </dd></dl>
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<p>Negates all the coefficients of the Quaternion.</p>
<p>This results in an other representation of the <em>same</em> rotation (opposite rotation angle, but with a negated axis direction: the two cancel out). However, note that the results of <a class="el" href="classqglviewer_1_1Quaternion.html#a63f3f585fd25e9cb32700a26d54f8ee4">axis()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a> are unchanged after a call to this method since <a class="el" href="classqglviewer_1_1Quaternion.html#a39bde10d142ce0cbc9614988ca0a0e59">angle()</a> always returns a value in [0,pi].</p>
<p>This method is mainly useful for Quaternion interpolation, so that the spherical interpolation takes the shortest path on the unit sphere. See <a class="el" href="classqglviewer_1_1Quaternion.html#a755201c9a752fd892a3656a517b190e4">slerp()</a> for details. </p>
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<p>Normalizes the Quaternion coefficients.</p>
<p>This method should not need to be called since we only deal with unit Quaternions. This is however useful to prevent numerical drifts, especially with small rotational increments. See also <a class="el" href="classqglviewer_1_1Quaternion.html#ae347ecf6e0a78618e8fd1a3b9107df32">normalized()</a>. </p>
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<p>Returns a normalized version of the Quaternion.</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a05826e509c686f39baaec4656f1a7231">normalize()</a>. </p>
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<p>Quaternion rotation is composed with <code>q</code>.</p>
<p>See <a class="el" href="classqglviewer_1_1Quaternion.html#a76b3ffdb188246ff6559069cb3f5b919">operator*()</a>, since this is equivalent to <code>this</code> = <code>this</code> * <code>q</code>.</p>
<dl class="note"><dt><b>Note:</b></dt><dd>For efficiency reasons, the resulting Quaternion is not normalized. You may <a class="el" href="classqglviewer_1_1Quaternion.html#a05826e509c686f39baaec4656f1a7231">normalize()</a> it after each application in case of numerical drift. </dd></dl>
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<p>Output stream operator. Enables debugging code like: </p>
<div class="fragment"><pre class="fragment"> <a class="code" href="classqglviewer_1_1Quaternion.html#a65ed15cc19af958b5933b5c522f10e66">Quaternion</a> rot(...);
cout << <span class="stringliteral">"Rotation="</span> << rot << endl;
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<p>Equal operator. </p>
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<p>Bracket operator returning an l-value. <code>i</code> must range in [0..3]. See the <a class="el" href="classqglviewer_1_1Quaternion.html#a81ba24ffd95778f3ca4e51b2b9922f45">Quaternion(double, double, double, double)</a> documentation. </p>
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<p>Bracket operator, with a constant return value. <code>i</code> must range in [0..3]. See the <a class="el" href="classqglviewer_1_1Quaternion.html#a81ba24ffd95778f3ca4e51b2b9922f45">Quaternion(double, double, double, double)</a> documentation. </p>
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<p>Returns a random unit Quaternion.</p>
<p>You can create a randomly directed unit vector using: </p>
<div class="fragment"><pre class="fragment">Vec randomDir = <a class="code" href="classqglviewer_1_1Quaternion.html#a87d5e98d958fdc63100979fcff1d9976">Quaternion::randomQuaternion</a>() * Vec(1.0, 0.0, 0.0); <span class="comment">// or any other Vec</span>
</pre></div><dl class="note"><dt><b>Note:</b></dt><dd>This function uses rand() to create pseudo-random numbers and the random number generator can be initialized using srand(). </dd></dl>
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<p>Returns the image of <code>v</code> by the Quaternion rotation.</p>
<p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a74d79bc1afcc02fdbfe0eb390c3546f5">inverseRotate()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a19ce6efe5ef2744c8293e8ba1a39b9e4">operator*(const Quaternion&, const Vec&)</a>. </p>
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<p>Sets the Quaternion as a rotation of axis <code>axis</code> and angle <code>angle</code> (in radians).</p>
<p><code>axis</code> does not need to be normalized. A null <code>axis</code> will result in an identity Quaternion. </p>
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<p>Sets the Quaternion from the three rotated vectors of an orthogonal basis.</p>
<p>The three vectors do not have to be normalized but must be orthogonal and direct (X^Y=k*Z, with k>0).</p>
<div class="fragment"><pre class="fragment"> <a class="code" href="classqglviewer_1_1Quaternion.html#a65ed15cc19af958b5933b5c522f10e66">Quaternion</a> q;
q.setFromRotatedBasis(X, Y, Z);
<span class="comment">// Now q.rotate(Vec(1,0,0)) == X and q.inverseRotate(X) == Vec(1,0,0)</span>
<span class="comment">// Same goes for Y and Z with Vec(0,1,0) and Vec(0,0,1).</span>
</pre></div><p>See also <a class="el" href="classqglviewer_1_1Quaternion.html#a1880f20e8080ee670b6a88ed0c7d69ac">setFromRotationMatrix()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a1b60be34a715145efc3b91e6dfba1634">Quaternion(const Vec&, const Vec&)</a>. </p>
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<p>Set the Quaternion from a (supposedly correct) 3x3 rotation matrix.</p>
<p>The matrix is expressed in European format: its three <em>columns</em> are the images by the rotation of the three vectors of an orthogonal basis. Note that OpenGL uses a symmetric representation for its matrices.</p>
<p><a class="el" href="classqglviewer_1_1Quaternion.html#a760617f082c950633b5642d2f60b2dd9">setFromRotatedBasis()</a> sets a Quaternion from the three axis of a rotated frame. It actually fills the three columns of a matrix with these rotated basis vectors and calls this method. </p>
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<p>Sets the Quaternion value. See the <a class="el" href="classqglviewer_1_1Quaternion.html#a81ba24ffd95778f3ca4e51b2b9922f45">Quaternion(double, double, double, double)</a> constructor documentation. </p>
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<td class="memname">Quaternion slerp </td>
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<td class="paramtype">const Quaternion & </td>
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<td class="paramtype">const Quaternion & </td>
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<td class="paramtype">float </td>
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<p>Returns the slerp interpolation of Quaternions <code>a</code> and <code>b</code>, at time <code>t</code>.</p>
<p><code>t</code> should range in [0,1]. Result is <code>a</code> when <code>t=0</code> and <code>b</code> when <code>t=1</code>.</p>
<p>When <code>allowFlip</code> is <code>true</code> (default) the slerp interpolation will always use the "shortest path" between the Quaternions' orientations, by "flipping" the source Quaternion if needed (see <a class="el" href="classqglviewer_1_1Quaternion.html#abcdb1512395327f8236a4f4a4d4ff648">negate()</a>). </p>
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<a class="anchor" id="a4b218b205d88a4dd50c8de38522b5812"></a><!-- doxytag: member="qglviewer::Quaternion::squad" ref="a4b218b205d88a4dd50c8de38522b5812" args="(const Quaternion &a, const Quaternion &tgA, const Quaternion &tgB, const Quaternion &b, float t)" -->
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<td class="paramtype">const Quaternion & </td>
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<p>Returns the slerp interpolation of the two Quaternions <code>a</code> and <code>b</code>, at time <code>t</code>, using tangents <code>tgA</code> and <code>tgB</code>.</p>
<p>The resulting Quaternion is "between" <code>a</code> and <code>b</code> (result is <code>a</code> when <code>t=0</code> and <code>b</code> for <code>t=1</code>).</p>
<p>Use <a class="el" href="classqglviewer_1_1Quaternion.html#a2a27d43a68da6c7c5243fe81e5a9d005">squadTangent()</a> to define the Quaternion tangents <code>tgA</code> and <code>tgB</code>. </p>
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<a class="anchor" id="a2a27d43a68da6c7c5243fe81e5a9d005"></a><!-- doxytag: member="qglviewer::Quaternion::squadTangent" ref="a2a27d43a68da6c7c5243fe81e5a9d005" args="(const Quaternion &before, const Quaternion &center, const Quaternion &after)" -->
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<td class="memname">Quaternion squadTangent </td>
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<td class="paramtype">const Quaternion & </td>
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<p>Returns a tangent Quaternion for <code>center</code>, defined by <code>before</code> and <code>after</code> Quaternions.</p>
<p>Useful for smooth spline interpolation of Quaternion with <a class="el" href="classqglviewer_1_1Quaternion.html#a4b218b205d88a4dd50c8de38522b5812">squad()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a755201c9a752fd892a3656a517b190e4">slerp()</a>. </p>
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<hr/><h2>Friends And Related Function Documentation</h2>
<a class="anchor" id="a19ce6efe5ef2744c8293e8ba1a39b9e4"></a><!-- doxytag: member="qglviewer::Quaternion::operator*" ref="a19ce6efe5ef2744c8293e8ba1a39b9e4" args="(const Quaternion &q, const Vec &v)" -->
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<td class="memname"><a class="el" href="classqglviewer_1_1Vec.html">Vec</a> operator* </td>
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<td class="paramname"> <em>v</em></td><td> </td>
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<p>Returns the image of <code>v</code> by the rotation <code>q</code>.</p>
<p>Same as q.rotate(v). See <a class="el" href="classqglviewer_1_1Quaternion.html#a5e2421069d88e576334e10f634625815">rotate()</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a74d79bc1afcc02fdbfe0eb390c3546f5">inverseRotate()</a>. </p>
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<a class="anchor" id="a76b3ffdb188246ff6559069cb3f5b919"></a><!-- doxytag: member="qglviewer::Quaternion::operator*" ref="a76b3ffdb188246ff6559069cb3f5b919" args="(const Quaternion &a, const Quaternion &b)" -->
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<td class="memname">Quaternion operator* </td>
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<td class="paramtype">const Quaternion & </td>
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<td class="paramname"> <em>b</em></td><td> </td>
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<p>Returns the composition of the <code>a</code> and <code>b</code> rotations.</p>
<p>The order is important. When applied to a <a class="el" href="classqglviewer_1_1Vec.html" title="The Vec class represents 3D positions and 3D vectors.">Vec</a> <code>v</code> (see <a class="el" href="classqglviewer_1_1Quaternion.html#a19ce6efe5ef2744c8293e8ba1a39b9e4">operator*(const Quaternion&, const Vec&)</a> and <a class="el" href="classqglviewer_1_1Quaternion.html#a5e2421069d88e576334e10f634625815">rotate()</a>) the resulting Quaternion acts as if <code>b</code> was applied first and then <code>a</code> was applied. This is obvious since the image <code>v'</code> of <code>v</code> by the composited rotation satisfies: </p>
<div class="fragment"><pre class="fragment"> v<span class="stringliteral">'= (a*b) * v = a * (b*v) </span>
</pre></div><p>Note that a*b usually differs from b*a.</p>
<dl class="attention"><dt><b>Attention:</b></dt><dd>For efficiency reasons, the resulting Quaternion is not normalized. Use <a class="el" href="classqglviewer_1_1Quaternion.html#a05826e509c686f39baaec4656f1a7231">normalize()</a> in case of numerical drift with small rotation composition. </dd></dl>
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