152 lines
6.8 KiB
TeX
152 lines
6.8 KiB
TeX
\documentclass[12pt, a4paper]{article}
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%packages
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\usepackage[ngerman]{babel}
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\usepackage[utf8x]{inputenc}
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%Formel packages
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\usepackage{amsmath}
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\usepackage{amsthm}
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\usepackage{amsbsy}
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\usepackage{amssymb}
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\usepackage{enumerate}
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\begin{document}
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\section*{Computer Graphics - Excercise 1}
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\subsection*{1.1.1}
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\subsection*{a)}
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\begin{itemize}
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\item Matrix Multiplication $\widehat{=}$ transformation
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\item Inverse of a transformation $\tilde{T}^{-1}$ \\
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\begin{align*}
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\Rightarrow && \tilde{T}*\tilde{T}^{-1} &= \tilde{T}^{-1}*\tilde{T} = \tilde{E} && \left |\ *\tilde{p} \right. \\
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&&\tilde{T}^{-1}*\tilde{T}*\tilde{p} &= \tilde{E}*\tilde{p} && \left |\ \tilde{T}*\tilde{p}=\tilde{p'} \right. \\
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\Rightarrow && \tilde{E}*\tilde{p} = \tilde{p}&, \hspace{1em} \tilde{T}^{-1}*\tilde{p'} = \tilde{p} \\
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\end{align*}
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\end{itemize}
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\subsection*{b)}
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For $R_1 \times R_2$\\
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\begin{gather*}
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\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}
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\end{gather*}
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For $R_2 \times R_1$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow R_1 \times R_2 \neq R_2 \times R_1 \Rightarrow$ does not commute\\
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\newpage
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\hspace{-2em}
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For $T_1 \times T_2$\\
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\begin{gather*}
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\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}
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\end{gather*}
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\hspace{-0.5em}
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For $T_2 \times T_1$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow T_1 \times T_2 = T_2 \times T_1 \Rightarrow$ does commute\\
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\\
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\hspace{-1.5em}
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For $S_1 \times S_2$\\
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\begin{gather*}
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\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}
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\end{gather*}
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For $S_2 \times S_1$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow S_1 \times S_2 = S_2 \times S_1 \Rightarrow$ does commute\\
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\\
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\hspace{-1.5em}
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For $R \times T$\\
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\begin{gather*}
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\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}
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\end{gather*}
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For $T \times R$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow R \times T \neq T \times R \Rightarrow$ does not commute\\
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\\
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\hspace{-1.5em}
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For $R \times S$\\
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\begin{gather*}
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\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}
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\end{gather*}
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For $S \times R$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow R \times S = S \times R \Rightarrow$ does commute\\
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\\
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\hspace{-1.5em}
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For $S \times T$\\
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\begin{gather*}
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\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}
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\end{gather*}
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For $T \times S$\\
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\begin{gather*}
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\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}
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\end{gather*}
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$\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\
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\subsection*{1.1.2}
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\subsection*{a)}
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\begin{gather*}
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\begin{pmatrix} 1 & 0 & 0 \\ 0&0&1\\ 0&1&0 \\ \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}
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\end{gather*}
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\subsection*{b)}
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\begin{gather*}
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\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} =
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\begin{pmatrix} \sum\limits_{i=1} x_i \\ \sum\limits_{i=1} y_i \\ \sum\limits_{i=1} z_i \end{pmatrix}
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\end{gather*}
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\subsection*{c)}
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\begin{gather*}
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\tilde{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};
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e = \begin{pmatrix} 5 \\ 10 \\ 5 \\\end{pmatrix}= t;
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p = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}
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\end{gather*}
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\begin{gather*}
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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\\
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y_\phi = \frac {\begin{pmatrix} 0\\1\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} cos(y_\phi) & 0 & sin(y_\phi) \\ 0 &1 &0\\ -sin(z_\phi)&0&cos(y_\phi) \\ \end{pmatrix}=R_y\\
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x_\phi = \frac {\begin{pmatrix} 1\\0\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} 0 & 0 & 0 \\ 0 &cos(x_\phi)&-sin(x_\phi)\\ 0&sin(x_\phi)&cos(x_\phi) \\ \end{pmatrix}=R_x\\
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M = R_x \times R_y \times R_z \\
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\end{gather*}
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\newpage
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Berechnung für: \\
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$p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
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\begin{gather*}
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\Rightarrow Vector =\begin{pmatrix}
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p_1 \times \tilde{M_1}\\
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p_2 \times \tilde{M_2}\\
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p_3 \times \tilde{M_3}\\
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p_4 \times \tilde{M_4}\\
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p_5 \times \tilde{M_5}\\ \end{pmatrix}
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=
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\begin{pmatrix}
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6.323746006808568\\
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2.672777625063053\\
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1.6528549605644152\\
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6.432049954608068\\
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4.731031538265404\\ \end{pmatrix}\\
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\Rightarrow z_{far} = p_4 = 6.432049954608068; z_{near} = p_3 = 1.6528549605644152\\
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\end{gather*}
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\end{document}
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