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Exc 1 edit theorie

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fruitstaa 2018-11-12 11:08:47 +01:00
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exercise1/Theorie.tex
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@ -122,14 +122,40 @@ $\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\
\subsection*{c)} \subsection*{c)}
\begin{align*} \begin{align*}
L(p) &= length(p_x - ((p_x \cdot (\vec{e} + \vec{d} * t)) *\vec{d} * t) - \vec{e}) \\ L(p) &= length(p_x - ((p_x \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \widehat{=} Anz. Punkte \\ Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \widehat{=} Anz. Punkte \\
\end{align*} \end{align*}
Berechnung: Berechnung für: \\
$p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
\begin{align*}
L(p_1) &= length(p_1 - ((p_1 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
L(p_2) &= length(p_2 - ((p_2 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
L(p_3) &= length(p_3 - ((p_3 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
L(p_4) &= length(p_4 - ((p_4 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
L(p_5) &= length(p_5 - ((p_5 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
\\
Vector &= ( L(p_1), L(p_2), L(p_3), L(p_4), L(p_5)) \\
\end{align*}
\begin{gather*}
M = \begin{pmatrix} M_{11} & M_{12} & M_{13} & t_x \\ M_{21}&M_{22}& M_{23}&t_y\\ M_{31}&M_{32}&M_{33}&t_z \\ p_x&p_y&p_z&1 \\ \end{pmatrix};
e = \begin{pmatrix} 5 \\ 10 \\ 5 \\\end{pmatrix}= t;
p = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}
\end{gather*}
\begin{gather*}
z_\phi = \frac {\begin{pmatrix} 0\\0\\1\\ \end{pmatrix} \times p_x} {1 \times p_x} \Rightarrow \begin{pmatrix} cos(z_\phi) & -sin(z_\phi) & 0 \\ sin(z_\phi)&cos(z_\phi)&0\\ 0&0&1 \\ \end{pmatrix}=R_z\\
y_\phi = \frac {\begin{pmatrix} 0\\0\\1\\ \end{pmatrix} \times p_x} {1 \times p_x} \Rightarrow \begin{pmatrix} cos(y_\phi) & -sin(y_\phi) & 0 \\ sin(y_\phi)&cos(y_\phi)&0\\ 0&0&1 \\ \end{pmatrix}=R_y\\
x_\phi = \frac {\begin{pmatrix} 0\\0\\1\\ \end{pmatrix} \times p_x} {1 \times p_x} \Rightarrow \begin{pmatrix} cos(x_\phi) & -sin(x_\phi) & 0 \\ sin(x_\phi)&cos(x_\phi)&0\\ 0&0&1 \\ \end{pmatrix}=R_x\\
M = R_x \times R_y \times R_z \\
\Rightarrow Vector = \begin{pmatrix}
6.323746006808568\\
2.672777625063053\\
1.6528549605644152\\
6.432049954608068\\
4.731031538265404\\ \end{pmatrix}\\
\Rightarrow z_{far} = p_4 = 6.432049954608068; z_{near} = p_3 = 1.6528549605644152\\
\end{gather*}
\end{document} \end{document}