%
% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\begin{frame}[t,fragile]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.44\textwidth}
\begin{block}{Determinante}
\[
A=\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}
\;\Rightarrow\;
\det A = ad-bc
\]
\end{block}
\end{column}
\begin{column}{0.52\textwidth}
\begin{block}{Dimension}
\[
4\; \text{Variablen}
-
1\; \text{Bedingung}
=
3\; \text{Dimensionen}
\]
\end{block}
\end{column}
\end{columns}
\vspace{-10pt}
\uncover<3->{%
\begin{columns}[t,onlytextwidth]
\def\s{0.94}
\begin{column}{0.33\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick,scale=\s]
\begin{scope}
	\clip (-2.1,-2.1) rectangle (2.3,2.3);
	\fill[color=blue!20] (-1,-1) rectangle (1,1);
	\foreach \x in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
	}
	\foreach \y in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
	}
	\ifthenelse{\boolean{presentation}}{
	\foreach \d in {4,...,10}{
		\only<\d>{
			\pgfmathparse{1+(\d-4)/10}
			\xdef\t{\pgfmathresult}
			\fill[color=red!40,opacity=0.5]
				({-\t},{-1/\t}) rectangle (\t,{1/\t});
			\foreach \x in {-2,...,2}{
				\draw[color=red,line width=0.3pt]
					({\x*\t},-3) -- ({\x*\t},3);
			}
			\foreach \y in {-3,...,3}{
				\draw[color=red,line width=0.3pt]
				(-3,{\y/\t}) -- (3,{\y/\t});
			}
		}
	}
	}{}
	\uncover<11->{
		\xdef\t{1.6}
		\fill[color=red!40,opacity=0.5]
			({-\t},{-1/\t}) rectangle (\t,{1/\t});
		\foreach \x in {-2,...,2}{
			\draw[color=red,line width=0.3pt]
				({\x*\t},-3) -- ({\x*\t},3);
		}
		\foreach \y in {-3,...,3}{
			\draw[color=red,line width=0.3pt]
			(-3,{\y/\t}) -- (3,{\y/\t});
		}
	}
\end{scope}
\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
\uncover<3->{%
	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
	\node at (0,-2.1) {$
	D
	=
	\begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
	$};
}
\end{tikzpicture}
\end{center}
\end{column}
\begin{column}{0.33\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick,scale=\s]
\fill[color=blue!20] (-1,-1) rectangle (1,1);
\begin{scope}
	\clip (-2.1,-2.1) rectangle (2.3,2.3);
	\foreach \x in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
	}
	\foreach \y in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
	}
	\ifthenelse{\boolean{presentation}}{
	\foreach \d in {11,...,17}{
		\only<\d>{
			\pgfmathparse{(\d-11)/10}
			\xdef\t{\pgfmathresult}
			\fill[color=red!40,opacity=0.5]
				({-1+\t*(-1)},{-1})
				--
				({1+\t*(-1)},{-1})
				--
				({1+\t},{1})
				--
				({-1+\t},{1})
				-- cycle;
			\foreach \x in {-3,...,3}{
				\draw[color=red,line width=0.3pt]
					({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
			}
			\foreach \y in {-3,...,3}{
				\draw[color=red,line width=0.3pt]
					({-3+\t*\y},\y) -- ({3+\t*\y},\y);
			}
		}
	}
	}{}
	\uncover<18->{
		\xdef\t{0.6}
		\fill[color=red!40,opacity=0.5]
			({-1+\t*(-1)},{-1})
			--
			({1+\t*(-1)},{-1})
			--
			({1+\t},{1})
			--
			({-1+\t},{1})
			-- cycle;
		\foreach \x in {-3,...,3}{
			\draw[color=red,line width=0.3pt]
				({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
		}
		\foreach \y in {-3,...,3}{
			\draw[color=red,line width=0.3pt]
				({-3+\t*\y},\y) -- ({3+\t*\y},\y);
		}
	}
\end{scope}
\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
\uncover<11->{
	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
	\node at (0,-2.1) {$
	S
	=
	\begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
	$};
}
\end{tikzpicture}
\end{center}
\end{column}
\begin{column}{0.33\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick,scale=\s]
\fill[color=blue!20] (-1,-1) rectangle (1,1);
\begin{scope}
	\clip (-2.1,-2.1) rectangle (2.3,2.3);
	\foreach \x in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
	}
	\foreach \y in {-2,...,2}{
		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
	}
	\ifthenelse{\boolean{presentation}}{
	\foreach \d in {18,...,24}{
		\only<\d>{
			\pgfmathparse{(\d-18)/10}
			\xdef\t{\pgfmathresult}
			\fill[color=red!40,opacity=0.5]
				(-1,{\t*(-1)-1})
				--
				(1,{\t*1-1})
				--
				(1,{\t*1+1})
				--
				(-1,{\t*(-1)+1})
				-- cycle;
			\foreach \x in {-3,...,3}{
				\draw[color=red,line width=0.3pt]
					(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
			}
			\foreach \y in {-3,...,3}{
				\draw[color=red,line width=0.3pt]
					(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
			}
		}
	}
	}{}
	\uncover<25->{
		\xdef\t{0.6}
		\fill[color=red!40,opacity=0.5]
			(-1,{\t*(-1)-1})
			--
			(1,{\t*1-1})
			--
			(1,{\t*1+1})
			--
			(-1,{\t*(-1)+1})
			-- cycle;
		\foreach \x in {-3,...,3}{
			\draw[color=red,line width=0.3pt]
				(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
		}
		\foreach \y in {-3,...,3}{
			\draw[color=red,line width=0.3pt]
				(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
		}
	}
\end{scope}
\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
\uncover<18->{%
\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
	\node at (0,-2.1) {$
	T
	=
	\begin{pmatrix} 1&0\\t&1\end{pmatrix}
	$};
}
\end{tikzpicture}
\end{center}
\end{column}
\end{columns}}
\end{frame}
\egroup