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b/exercise1/Theorie.tex new file mode 100644 index 0000000..97fdbbb --- /dev/null +++ b/exercise1/Theorie.tex @@ -0,0 +1,55 @@ +\documentclass[12pt, a4paper]{scrbook} + +%packages +\usepackage[ngerman]{babel} +\usepackage[utf8x]{inputenc} +%Formel packages +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amsbsy} +\usepackage{amssymb} + + +\usepackage{enumerate} + + +\begin{document} + +\section*{Computer Graphics - Excercise 1} +\subsection*{1.1.1} +\subsection*{a)} +\begin{itemize} +\item Matrix Multiplication $\equiv$ 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)} + +\subsection*{1.1.2} +\subsection*{a)} +\begin{gather*} +\begin{pmatrix} 1 & 0 & 0 \\ 0&1&0\\ 0&0&1 \\ \end{pmatrix} * \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} * \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \\ \end{pmatrix} = +\begin{pmatrix} \sum\limits_{i=1} x \\ \sum\limits_{i=1} y \\ \sum\limits_{i=1} z \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 \equiv Anz. Punkte \\ +\end{align*} + +Berechnung: + + + + +\end{document} +