hpr.tex

BCView - Bayes Classifier Visualization Download xbcview Linux executable (218 kb) wbcview.exe W

\documentclass{article}
\usepackage{german}
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\textwidth     155mm
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\textheight    230mm

\def\rgtbox#1#2{\phantom{#1}\hbox to0pt{\hss #2}}


\begin{document}

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\subsubsection*{Computation of the View Coordinate System}

Rotation matrices for the individual axes:
\begin{center}
\begin{tabular}{@{}c@{\qquad}c@{\qquad}c@{}}
Heading & Pitch & Roll \\
rotation around $z$-axis &
rotation around $x$-axis &
rotation around $y$-axis \\[1ex]
$\displaystyle {\mathbf H} =
\left(\begin{array}{rrr}
\cos h & -\sin h &       0 \\
\sin h &  \cos h &       0 \\
     0 &       0 &       1
\end{array}\right)$ &
$\displaystyle {\mathbf P} =
\left(\begin{array}{rrr}
     1 &       0 &       0 \\
     0 &  \cos p & -\sin p \\
     0 &  \sin p &  \cos p
\end{array}\right)$ &
$\displaystyle {\mathbf R} =
\left(\begin{array}{rrr}
\cos r &       0 & -\sin r \\
     0 &       1 &       0 \\
\sin r &       0 &  \cos r
\end{array}\right)$
\end{tabular}
\end{center}
Combination of the matrices for the individual axes: \quad
${\mathbf M} = {\mathbf H} \cdot {\mathbf P} \cdot {\mathbf R}$. \\[1ex]
First step:
Compute ${\mathbf H} \cdot {\mathbf P}$.
\begin{center}
\begin{tabular}{@{}cl@{}}
&
$\displaystyle
\left(\begin{array}{rrr}
\rgtbox{\cos h}{1} & 0                       & 0 \\
0                  & \cos p                  & \phantom{\cos h}-\sin p\\
0                  & \phantom{-\sin h}\sin p & \cos p
\end{array}\right)$ \\[4ex]
$\displaystyle
\left(\begin{array}{rrr}
\cos h & -\sin h & 0 \\
\sin h &  \cos h & 0 \\
0      &  0      & 1
\end{array}\right)$ &
$\displaystyle
\left(\begin{array}{rrr}
\cos h & -\sin h \cos p &  \sin h \sin p \\
\sin h &  \cos h \cos p & -\cos h \sin p \\
0      &  \sin p        &  \cos p
\end{array}\right)$
\end{tabular}
\end{center}
Second step:
Compute $({\mathbf H} \cdot {\mathbf P}) \cdot {\mathbf R}$.
\begin{center}
\begin{tabular}{@{}l@{}}
$\displaystyle
\left(\begin{array}{rrr}
\cos r & 0 & \rgtbox{-\cos h\sin r +\sin h\sin p\cos r}{$-\sin r$}\\
0      & \rgtbox{-\sin h \cos p}{1}            &       0\\
\phantom{\cos h\cos r +\sin h\sin p}\sin r & 0 &  \cos r
\end{array}\right)$ \\[4ex]
$\displaystyle
\left(\begin{array}{rrr}
\cos h\cos r +\sin h\sin p\sin r &
   -\sin h\cos p & -\cos h\sin r +\sin h\sin p\cos r\\
\sin h\cos r -\cos h\sin p\sin r &
    \cos h\cos p & -\sin h\sin r -\cos h\sin p\cos r\\
\cos p\sin r                     &
    \sin p       &  \cos p\cos r
\end{array}\right)$
\end{tabular}
\end{center}
Consequently, the axes of the view coordinate system are:
\begin{eqnarray*}
\vec{v}_x & = & \left(\begin{array}{r}
                 \cos h\cos p +\sin h\sin p\sin r \\
                 \sin h\cos r -\cos h\sin p\sin r \\
                 \cos p\sin r
                \end{array}\right) \\
\vec{v}_y & = & \left(\begin{array}{r}
                -\sin h\cos p \\
                 \cos h\cos p \\
                 \sin p
                \end{array}\right) \\
\vec{v}_z & = & \left(\begin{array}{r}
                -\cos h\sin r +\sin h\sin p\cos r\\
                -\sin h\sin r -\cos h\sin p\cos r\\
                 \cos p\cos r
                \end{array}\right)
\end{eqnarray*}
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\end{document}