minor

Author: ralfHielscher

pages/documentation_matlab/SO3Kernels.html

@@ -22,7 +22,7 @@
   bandwidth: 3
   halfwidth: 40°
 {% endhighlight %}
-<p>We plot this function by evaluation of its Chebychev series in \(\cos(\frac{\omega}{2})\) for \(\omega\in[-pi,\pi].\)</p>
+<p>We plot this function by evaluation of its Chebychev series in \(\cos(\frac{\omega}{2})\) for \(\omega \in [-\pi,\pi]\).</p>
 {% highlight matlab %}
 plot(psi)
 {% endhighlight %}
@@ -49,7 +49,7 @@
 
   Bunge Euler angles in degree
      phi1     Phi    phi2  weight
-  345.959 150.052 302.531       1
+  42.7367 170.109 84.2033       1
 {% endhighlight %}
 <center>
 {% include inline_image.html file="SO3Kernels_02.png" %}
@@ -326,7 +326,7 @@
 {% endhighlight %}
 <center>
 {% include inline_image.html file="SO3Kernels_19.png" %}
-</center><h2 id="24">The Bump kernel</h2><p>The <a href="SO3Kernels.SO3BumpKernel.html">bump kernel</a> \(\tilde\psi_r\in L^2(\mathcal{SO}(3))\) is a radial symmetric kernel function depending on a parameter \(r\in (0,pi)\). The function value is 0, if the angle is greater then the halfwidth \(r\). Otherwise it is has a contstant value, such that the mean of \(\psi_r\) on \(\mathcal{SO}(3)\) is 1. Hence we use the open set</p><p>\[U_r = \{ \bf{R} \in \mathcal{SO}(3) \,\vert ~ \lvert \omega( \bf{R})\rvert &lt;r \}\]</p><p>and define the bump kernel by</p><p>\[ \tilde\psi_r( \bf{R}) = \frac1{\lvert U_r \rvert } \mathbf{1}_{ \{ \bf{R} \in U_r \} } \]</p><p>where \(\mathbf{1}\) is the indicator function.</p><p>The main problem of the bump kernel is that we need a lot of chebychev coefficients to describe it. That possibly can result in high runtimes.</p>
+</center><h2 id="24">The Bump kernel</h2><p>The <a href="SO3Kernels.SO3BumpKernel.html">bump kernel</a> \(\tilde\psi_r\in L^2(\mathcal{SO}(3))\) is a radial symmetric kernel function depending on a parameter \(r\in (0,\pi)\). The function value is 0, if the angle is greater then the halfwidth \(r\). Otherwise it is has a contstant value, such that the mean of \(\psi_r\) on \(\mathcal{SO}(3)\) is 1. Hence we use the open set</p><p>\[U_r = \{ \bf{R} \in \mathcal{SO}(3) \,\vert ~ \lvert \omega( \bf{R})\rvert &lt;r \}\]</p><p>and define the bump kernel by</p><p>\[ \tilde\psi_r( \bf{R}) = \frac1{\lvert U_r \rvert } \mathbf{1}_{ \{ \bf{R} \in U_r \} } \]</p><p>where \(\mathbf{1}\) is the indicator function.</p><p>The main problem of the bump kernel is that we need a lot of chebychev coefficients to describe it. That possibly can result in high runtimes.</p>
 {% highlight matlab %}
 psi1 = SO3BumpKernel(30*degree)
 psi2 = SO3BumpKernel(40*degree)