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<article class="doc">
<h1 class="page">Matrix</h1>
<div class="sect1">
<h2 id="matrix"><a class="anchor" href="#matrix"></a>Matrix</h2>
<div class="sectionbody">
<div class="paragraph">
<p>See <a href="https://javadoc.jmonkeyengine.org/v3.4.1-stable/com/jme3/math/Matrix3f.html">Javadoc of Matrix3f</a> and <a href="https://javadoc.jmonkeyengine.org/v3.4.1-stable/com/jme3/math/Matrix4f.html">Javadoc of Matrix4f</a></p>
</div>
<div class="sect2">
<h3 id="definition"><a class="anchor" href="#definition"></a>Definition</h3>
<div class="paragraph">
<p>A Matrix is typically used as a <em>linear transformation</em> to map vectors to vectors. That is: Y = MX where X is a Vector and M is a Matrix applying any or all transformations (scale, rotate, translate).</p>
</div>
<div class="paragraph">
<p>There are a few special matrices:</p>
</div>
<div class="paragraph">
<p><em>zero matrix</em> is the Matrix with all zero entries.</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 33.3333%;">
<col style="width: 33.3333%;">
<col style="width: 33.3334%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">0</th>
<th class="tableblock halign-left valign-top">0</th>
<th class="tableblock halign-left valign-top">0</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>The <em>Identity Matrix</em> is the matrix with 1 on the diagonal entries and 0 for all other entries.</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 33.3333%;">
<col style="width: 33.3333%;">
<col style="width: 33.3334%;">
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<thead>
<tr>
<th class="tableblock halign-left valign-top">1</th>
<th class="tableblock halign-left valign-top">0</th>
<th class="tableblock halign-left valign-top">0</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>1</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>1</p>
</div></div></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>A Matrix is <em>invertible</em> if there is a matrix <em>M<sup>-1</sup></em> where <em>MM<sup>-1</sup> = M<sup>-1</sup> = I</em>.</p>
</div>
<div class="paragraph">
<p>The <em>transpose</em> of a matrix <em>M = [m<sub>ij</sub>]</em> is <em>M<sup>T</sup> = [m<sub>ji</sub>]</em>. This causes the rows of <em>M</em> to become the columns of <em>M<sup>T</sup></em>.</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2858%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">1</th>
<th class="tableblock halign-left valign-top">1</th>
<th class="tableblock halign-left valign-top">1</th>
<th class="tableblock halign-left valign-top"></th>
<th class="tableblock halign-left valign-top">1</th>
<th class="tableblock halign-left valign-top">2</th>
<th class="tableblock halign-left valign-top">3</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>2</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>2</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>2</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>⇒</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>1</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>2</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>3</p>
</div></div></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>3</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>3</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>3</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>1</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>2</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>3</p>
</div></div></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>A Matrix is symmetric if <em>M</em> = <em>M<sup>T</sup></em>.</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 33.3333%;">
<col style="width: 33.3333%;">
<col style="width: 33.3334%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">X</th>
<th class="tableblock halign-left valign-top">A</th>
<th class="tableblock halign-left valign-top">B</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>A</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>X</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>C</p>
</div></div></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>B</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>C</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>X</p>
</div></div></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>Where X, A, B, and C equal numbers</p>
</div>
<div class="paragraph">
<p>jME includes two types of Matrix classes: Matrix3f and Matrix4f. Matrix3f is a 3x3 matrix and is the most commonly used (able to handle scaling and rotating), while Matrix4f is a 4x4 matrix that can also handle translation.</p>
</div>
</div>
<div class="sect2">
<h3 id="transformations"><a class="anchor" href="#transformations"></a>Transformations</h3>
<div class="paragraph">
<p>Multiplying a <a href="../../tutorials/concepts/terminology.html#vectors" class="xref page">Vector</a> with a Matrix allows the Vector to be transformed. Either rotating, scaling or translating that Vector.</p>
</div>
<div class="sect3">
<h4 id="scaling"><a class="anchor" href="#scaling"></a>Scaling</h4>
<div class="paragraph">
<p>If a <em>diagonal Matrix</em>, defined by D = [d<sub>ij</sub>] and d<sub>ij</sub> = 0 for i != j, has all positive entries it is a <em>scaling matrix</em>. If d<sub>i</sub> is greater than 1 then the resulting Vector will grow, while if d<sub>i</sub> is less than 1 it will shrink.</p>
</div>
</div>
<div class="sect3">
<h4 id="rotation"><a class="anchor" href="#rotation"></a>Rotation</h4>
<div class="paragraph">
<p>A <em>rotation matrix</em> requires that the transpose and inverse are the same matrix (R<sup>-1</sup> = R<sup>T</sup>). The <em>rotation matrix</em> R can then be calculated as: R = I + (sin(angle)) S + (1 - cos(angle)S<sup>2</sup> where S is:</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 33.3333%;">
<col style="width: 33.3333%;">
<col style="width: 33.3334%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">0</th>
<th class="tableblock halign-left valign-top">u<sub>2</sub></th>
<th class="tableblock halign-left valign-top">-u<sub>1</sub></th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>-u<sub>2</sub></p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>u<sub>0</sub></p>
</div></div></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>u<sub>1</sub></p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>-u<sub>0</sub></p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>0</p>
</div></div></td>
</tr>
</tbody>
</table>
</div>
<div class="sect3">
<h4 id="translation"><a class="anchor" href="#translation"></a>Translation</h4>
<div class="paragraph">
<p>Translation requires a 4x4 matrix, where the Vector (x,y,z) is mapped to (x,y,z,1) for multiplication. The <em>Translation Matrix</em> is then defined as:</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 50%;">
<col style="width: 50%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">M</th>
<th class="tableblock halign-left valign-top">T</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>S<sup>T</sup></p>
</div></div></td>
<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
<p>1</p>
</div></div></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>where M is the 3x3 matrix (containing any rotation/scale information), T is the translation Vector and S<sup>T</sup> is the transpose Vector of T. 1 is just a constant.</p>
</div>
</div>
</div>
<div class="sect2">
<h3 id="jme-class"><a class="anchor" href="#jme-class"></a>jME Class</h3>
<div class="paragraph">
<p>Both Matrix3f and Matrix4f store their values as floats and are publicly available as (m00, m01, m02, …, mNN) where N is either 2 or 3.</p>
</div>
<div class="paragraph">
<p>Most methods are straight forward, and I will leave documentation to the Javadoc.</p>
</div>
</div>
</div>
</div>
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