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<span id="Vector-Rotation-Matrices"></span><div class="header">
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Previous: <a href="Interpolation-on-Scattered-Data.html" accesskey="p" rel="prev">Interpolation on Scattered Data</a>, Up: <a href="Geometry.html" accesskey="u" rel="up">Geometry</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
<span id="Vector-Rotation-Matrices-1"></span><h3 class="section">30.5 Vector Rotation Matrices</h3>
<p>Also included in Octave’s geometry functions are primitive functions to enable
vector rotations in 3-dimensional space. Separate functions are provided for
rotation about each of the principle axes, <var>x</var>, <var>y</var>, and <var>z</var>.
According to Euler’s rotation theorem, any arbitrary rotation, <var>R</var>, of any
vector, <var>p</var>, can be expressed as a product of the three principle
rotations:
</p>
<div class="example">
<pre class="example">p' = Rp = Rz*Ry*Rx*p
</pre></div>
<span id="XREFrotx"></span><dl>
<dt id="index-rotx">: <em><var>T</var> =</em> <strong>rotx</strong> <em>(<var>angle</var>)</em></dt>
<dd>
<p><code>rotx</code> returns the 3x3 transformation matrix corresponding to an active
rotation of the vector about the x-axis by the specified <var>angle</var>, given
in degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the y-z plane from the positive x side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<pre class="example"> | 1 0 0 |
T = | 0 cos(<var>angle</var>) -sin(<var>angle</var>) |
| 0 sin(<var>angle</var>) cos(<var>angle</var>) |
</pre></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation <var>v</var> = <var>T</var><var>u</var>.
For example, a vector, <var>u</var>, pointing along the positive y-axis, rotated
90-degrees about the x-axis, will result in a vector pointing along the
positive z-axis:
</p>
<div class="example">
<pre class="example">>> u = [0 1 0]'
u =
0
1
0
>> T = rotx (90)
T =
1.00000 0.00000 0.00000
0.00000 0.00000 -1.00000
0.00000 1.00000 0.00000
>> v = T*u
v =
0.00000
0.00000
1.00000
</pre></div>
<p><strong>See also:</strong> <a href="#XREFroty">roty</a>, <a href="#XREFrotz">rotz</a>.
</p></dd></dl>
<span id="XREFroty"></span><dl>
<dt id="index-roty">: <em><var>T</var> =</em> <strong>roty</strong> <em>(<var>angle</var>)</em></dt>
<dd>
<p><code>roty</code> returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the y-axis by the specified <var>angle</var>, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the z-x plane from the positive y side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<pre class="example"> | cos(<var>angle</var>) 0 sin(<var>angle</var>) |
T = | 0 1 0 |
| -sin(<var>angle</var>) 0 cos(<var>angle</var>) |
</pre></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation <var>v</var> = <var>T</var><var>u</var>.
For example, a vector, <var>u</var>, pointing along the positive z-axis, rotated
90-degrees about the y-axis, will result in a vector pointing along the
positive x-axis:
</p>
<div class="example">
<pre class="example"> >> u = [0 0 1]'
u =
0
0
1
>> T = roty (90)
T =
0.00000 0.00000 1.00000
0.00000 1.00000 0.00000
-1.00000 0.00000 0.00000
>> v = T*u
v =
1.00000
0.00000
0.00000
</pre></div>
<p><strong>See also:</strong> <a href="#XREFrotx">rotx</a>, <a href="#XREFrotz">rotz</a>.
</p></dd></dl>
<span id="XREFrotz"></span><dl>
<dt id="index-rotz">: <em><var>T</var> =</em> <strong>rotz</strong> <em>(<var>angle</var>)</em></dt>
<dd>
<p><code>rotz</code> returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the z-axis by the specified <var>angle</var>, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the x-y plane from the positive z side.
</p>
<p>The form of the transformation matrix is:
</p>
<div class="example">
<pre class="example"> | cos(<var>angle</var>) -sin(<var>angle</var>) 0 |
T = | sin(<var>angle</var>) cos(<var>angle</var>) 0 |
| 0 0 1 |
</pre></div>
<p>This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation <var>v</var> = <var>T</var><var>u</var>.
For example, a vector, <var>u</var>, pointing along the positive x-axis, rotated
90-degrees about the z-axis, will result in a vector pointing along the
positive y-axis:
</p>
<div class="example">
<pre class="example"> >> u = [1 0 0]'
u =
1
0
0
>> T = rotz (90)
T =
0.00000 -1.00000 0.00000
1.00000 0.00000 0.00000
0.00000 0.00000 1.00000
>> v = T*u
v =
0.00000
1.00000
0.00000
</pre></div>
<p><strong>See also:</strong> <a href="#XREFrotx">rotx</a>, <a href="#XREFroty">roty</a>.
</p></dd></dl>
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