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\begin{document}

\section*{Computer Graphics - Excercise 1}
\subsection*{1.1.1}
\subsection*{a)}
\begin{itemize}
\item		Matrix Multiplication $\widehat{=}$ transformation
\item		Inverse of a transformation $\tilde{T}^{-1}$ \\
\begin{align*}
		\Rightarrow && \tilde{T}*\tilde{T}^{-1} &= \tilde{T}^{-1}*\tilde{T} = \tilde{E} && \left |\ *\tilde{p} \right. \\
		&&\tilde{T}^{-1}*\tilde{T}*\tilde{p} &= \tilde{E}*\tilde{p} && \left |\ \tilde{T}*\tilde{p}=\tilde{p'} \right. \\
		\Rightarrow && \tilde{E}*\tilde{p} = \tilde{p}&, \hspace{1em} \tilde{T}^{-1}*\tilde{p'} = \tilde{p} \\
\end{align*}
\end{itemize}
\subsection*{b)}

For $R_1 \times R_2$\\
\begin{gather*}
\begin{pmatrix} a & b & 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} e & f & 0 \\  g& h& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times e + b \times g& a \times f + b \times h & 0 \\  c \times e + d \times g& c \times f + d \times h & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
For $R_2 \times R_1$\\
\begin{gather*}
\begin{pmatrix} e & f & 0 \\  g& h& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b & 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix}  =\begin{pmatrix} a \times e + c \times f& b \times e + d \times f & 0 \\  a \times g + c \times h& b \times g + d \times h & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow R_1 \times R_2 \neq R_2 \times R_1 \Rightarrow$ does not commute\\

\newpage

\hspace{-5.5em}
For $T_1 \times T_2$\\
\begin{gather*}
\begin{pmatrix} 1 & 0 & a  \\ 0&1&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & c \\  0& 1& d\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix}  1& 0 & a+c \\  0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
\hspace{-4.5em}
For $T_2 \times T_1$\\
\begin{gather*}
\begin{pmatrix} 1 & 0 & c  \\ 0&1&d\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a \\  0& 1& b\\ 0&0&1\\ \end{pmatrix}  =\begin{pmatrix}  1& 0 & a+c \\  0& 1 & b+d\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow T_1 \times T_2 = T_2 \times T_1 \Rightarrow$ does commute\\

\hspace{-5.5em}
For $S_1 \times S_2$\\
\begin{gather*}
\begin{pmatrix} a & 0 & 0  \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\  0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
\hspace{-4.5em}
For $S_2 \times S_1$\\
\begin{gather*}
\begin{pmatrix} c & 0& 0 \\  0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & 0 & 0  \\ 0&b&0\\ 0&0&1 \\ \end{pmatrix}  =\begin{pmatrix} a \times c & 0 & 0 \\ b \times d& 0 & 0\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow S_1 \times S_2 = S_2 \times S_1 \Rightarrow$ does commute\\

\hspace{-5.5em}
For $R \times T$\\
\begin{gather*}
\begin{pmatrix} a & b& 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & e \\  0& 1& f\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix}  a& b & a \times e + b \times f \\  c& d & c\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
\hspace{-4.5em}
For $T \times R$\\
\begin{gather*}
\begin{pmatrix} 1 & 0 & e \\  0& 1& f\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix}  =\begin{pmatrix}  a& b & e \\  c& d & f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow R \times T \neq T \times R \Rightarrow$ does not commute\\

\hspace{-1.25em}
For $R \times S$\\
\begin{gather*}
\begin{pmatrix} a & b& 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} e & 0 & 0 \\  0& f& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix}  a \times e& b \times f & 0 \\  c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
\hspace{-0.25em}
For $S \times R$\\
\begin{gather*}
\begin{pmatrix} e & 0 & 0 \\  0& f& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} a & b& 0  \\ c&d&0\\ 0&0&1 \\ \end{pmatrix}  =\begin{pmatrix}  a \times e& b \times f & 0 \\  c \times f & d \times f & 0\times e + d \times f\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow R \times S = S \times R \Rightarrow$ does commute\\

\hspace{-1.25em}
For $S \times T$\\
\begin{gather*}
\begin{pmatrix} 1 & 0 & a  \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} c & 0 & 0 \\  0& d& 0\\ 0&0&1\\ \end{pmatrix} =\begin{pmatrix} c & 0 & a \\ 0& d & b\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}
\hspace{-0.25em}
For $T \times S$\\
\begin{gather*}
\begin{pmatrix} c & 0 & 0 \\  0& d& 0\\ 0&0&1\\ \end{pmatrix} \times \begin{pmatrix} 1 & 0 & a  \\ 0&0&b\\ 0&0&1 \\ \end{pmatrix}  =\begin{pmatrix} c & 0 & a \times c \\ d& 0 & b \times d\\ 0&0&1 \\ \end{pmatrix}
\end{gather*}

$\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\


\subsection*{1.1.2}
\subsection*{a)}
\begin{gather*}
\begin{pmatrix} 1 & 0 & 0  \\ 0&1&0\\ 0&0&1 \\ \end{pmatrix} \times \begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ y_1&y_2&\cdots&y_n\\ z_1&z_2&\cdots&z_n \\ \end{pmatrix} =\begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ z_1&z_2&\cdots&z_n\\ y_1&y_2&\cdots&y_n \\ \end{pmatrix}
\end{gather*}

\subsection*{b)}
\begin{gather*}
\begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ y_1&y_2&\cdots&y_n\\ z_1&z_2&\cdots&z_n \\ \end{pmatrix} \times \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \\ \end{pmatrix} = 
\begin{pmatrix}  \sum\limits_{i=1} x_i \\ \sum\limits_{i=1} y_i \\ \sum\limits_{i=1} z_i \end{pmatrix}
\end{gather*}
\subsection*{c)}

\begin{align*}
L(p) &= length(p_x - ((p_x \cdot (\vec{e} + \vec{d} * t)) *\vec{d} * t) - \vec{e}) \\
Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \widehat{=} Anz. Punkte \\
\end{align*}

Berechnung:




\end{document}