simpliziale Approximation

10 files changed, 181 insertions, 9 deletions

diff --git a/buch/chapters/95-homologie/images/Makefile b/buch/chapters/95-homologie/images/Makefile
index 50c2b0d..bc85c55 100644
--- a/buch/chapters/95-homologie/images/Makefile
+++ b/buch/chapters/95-homologie/images/Makefile
@@ -4,7 +4,8 @@
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
 all:	complexbasis.pdf homocycles.pdf homoboundaries.pdf homoclasses.pdf \
-	gausshomoex.pdf gausshomobasis.pdf dreieck.pdf polyeder.pdf
+	gausshomoex.pdf gausshomobasis.pdf dreieck.pdf polyeder.pdf \
+	approximation.pdf
 
 dreieck.pdf:	dreieck.tex
 	pdflatex dreieck.tex
@@ -30,3 +31,9 @@ homoclasses.pdf:	homoclasses.tex
 complexbasis.pdf:	complexbasis.tex
 	pdflatex complexbasis.tex
 
+approximation.pdf:	approximation.tex approx.tex
+	pdflatex approximation.tex
+
+approx.tex:	approx.m
+	octave approx.m
+
diff --git a/buch/chapters/95-homologie/images/approx.m b/buch/chapters/95-homologie/images/approx.m
new file mode 100644
index 0000000..0db41c2
--- /dev/null
+++ b/buch/chapters/95-homologie/images/approx.m
@@ -0,0 +1,77 @@
+#
+# approx.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+x = zeros(7,7);
+y = zeros(7,7);
+
+s = 1.05;
+
+for i = (1:7)
+	winkel = (i-1) * 8.333333 + 20;
+	for j = (1:7)
+		radius = (j-1) * 0.5 + 3;
+		x(i,j) = 1.05 * radius * cosd(winkel);
+		y(i,j) = 1.05 * radius * sind(winkel);
+	endfor
+endfor
+
+X = x;
+Y = y;
+for i = (1:7)
+	for j = (1:7)
+		X(i,j) = round(2 * x(i,j)) / 2;
+		Y(i,j) = round(2 * y(i,j)) / 2;
+	endfor
+endfor
+
+fn = fopen("approx.tex", "w");
+
+
+for i = (1:6)
+	for j = (1:6)
+		winkel = (i-1+0.6666) * 8.33333 + 20;
+		radius = (j-1+0.3333) * 0.5 + 3;
+		fprintf(fn, "\\definecolor{mycolor}{rgb}{%.2f,%.2f,%.2f};\n",
+			(winkel - 20) / 50, 0.8, (radius-3)/3);
+		fprintf(fn, "\\fill[color=mycolor] (%.3f,%.3f) -- (%.3f,%.3f) -- (%.3f,%.3f) -- cycle;\n",
+			X(i,j), Y(i,j),
+			X(i+1,j+1), Y(i+1,j+1),
+			X(i+1,j), Y(i+1,j));
+		winkel = (i-1+0.3333) * 8.33333 + 20;
+		radius = (j-1+0.6666) * 0.5 + 3;
+		fprintf(fn, "\\definecolor{mycolor}{rgb}{%.2f,%.2f,%.2f};\n",
+			(winkel - 20) / 50, 0.8, (radius-3)/3);
+		fprintf(fn, "\\fill[color=mycolor] (%.3f,%.3f) -- (%.3f,%.3f) -- (%.3f,%.3f) -- cycle;\n",
+			X(i,j), Y(i,j),
+			X(i,j+1), Y(i,j+1),
+			X(i+1,j+1), Y(i+1,j+1));
+	endfor
+endfor
+
+linewidth = 0.4;
+
+fprintf(fn, "\\gitter\n");
+	
+for i = (1:7)
+	for j = (1:6)
+		fprintf(fn, "\\draw[color=darkred,line width=%.1fpt] (%.3f,%.3f) -- (%.3f,%.3f);\n", linewidth,
+			X(i,j), Y(i,j), X(i,j+1), Y(i,j+1));
+	endfor
+endfor
+for i = (1:6)
+	for j = (1:7)
+		fprintf(fn, "\\draw[color=darkred,line width=%.1fpt] (%.3f,%.3f) -- (%.3f,%.3f);\n", linewidth,
+			X(i,j), Y(i,j), X(i+1,j), Y(i+1,j));
+	endfor
+endfor
+for i = (1:6)
+	for j = (1:6)
+		fprintf(fn, "\\draw[color=darkred,line width=%.1fpt] (%.3f,%.3f) -- (%.3f,%.3f);\n", linewidth,
+			X(i,j), Y(i,j), X(i+1,j+1), Y(i+1,j+1));
+	endfor
+endfor
+
+fclose(fn)
+
diff --git a/buch/chapters/95-homologie/images/approximation.pdf b/buch/chapters/95-homologie/images/approximation.pdf
new file mode 100644
index 0000000..8bdd2e7
--- /dev/null
+++ b/buch/chapters/95-homologie/images/approximation.pdf
diff --git a/buch/chapters/95-homologie/images/approximation.tex b/buch/chapters/95-homologie/images/approximation.tex
new file mode 100644
index 0000000..042f0e2
--- /dev/null
+++ b/buch/chapters/95-homologie/images/approximation.tex
@@ -0,0 +1,69 @@
+%
+% approximation.tex -- Approximation einer Abbildung durch eine simpliziale
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1.3}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkred}{rgb}{0.8,0,0}
+\definecolor{darkred}{rgb}{0,0,0}
+
+\def\gitter{
+	\foreach \x in {1,1.5,...,6}{
+		\draw[color=gray] (\x,1) -- (\x,6);
+		\draw[color=gray] (1,\x) -- (6,\x);
+	}
+}
+
+\def\s{1.05}
+
+\def\colorsector{
+	\foreach \r in {3,3.2,...,5.8}{
+		\foreach \a in {20,...,69}{
+			\pgfmathparse{(\a-20)/50}
+			\xdef\rot{\pgfmathresult}
+			\pgfmathparse{(\r-3)/3}
+			\xdef\blau{\pgfmathresult}
+			\definecolor{mycolor}{rgb}{\rot,0.8,\blau}
+			\fill[color=mycolor]
+				(\a:{\s*\r}) -- (\a:{\s*(\r+0.2)}) -- ({\a+1}:{\s*(\r+0.2)}) -- ({\a+1}:{\s*\r}) -- cycle;
+		}
+	}
+}
+
+\begin{scope}[xshift=0cm]
+\colorsector
+\gitter
+\foreach \r in {3,3.5,...,6.0}{
+	\draw[color=black,line width=0.4pt] (20:{\s*\r}) arc (20:70:{\s*\r});
+}
+\foreach \a in {20,28.3333,...,70}{
+	\draw[color=black,line width=0.4pt] (\a:{\s*3}) -- (\a:{\s*6});
+}
+\begin{scope}
+\clip (20:{\s*3}) -- (20:{\s*6}) arc (20:70:{\s*6}) -- (70:{\s*3});
+\foreach \a in {-5,...,5}{
+	\draw[color=black,line width=0.4pt]
+		plot[domain={20+8.33333*\a}:{70+8.3333*\a},samples=100]
+			(\x:{\s*(3+3*(\x-(20+8.3333*\a))/50)});
+}
+\end{scope}
+
+\end{scope}
+
+\begin{scope}[xshift=5.5cm]
+\input{approx.tex}
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/95-homologie/images/homoboundaries.pdf b/buch/chapters/95-homologie/images/homoboundaries.pdf
index 644f334..fb94ec8 100644
--- a/buch/chapters/95-homologie/images/homoboundaries.pdf
+++ b/buch/chapters/95-homologie/images/homoboundaries.pdf
diff --git a/buch/chapters/95-homologie/images/homoboundaries.tex b/buch/chapters/95-homologie/images/homoboundaries.tex
index ef8fd1a..53087fa 100644
--- a/buch/chapters/95-homologie/images/homoboundaries.tex
+++ b/buch/chapters/95-homologie/images/homoboundaries.tex
@@ -14,7 +14,7 @@
 \def\skala{1}
 \begin{tikzpicture}[>=latex,thick,scale=\skala]
 
-\def\s{0.4}
+\def\s{0.55}
 
 \def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
 \def\A{\punkt{0}{0}}
@@ -41,6 +41,7 @@
 \def\blau#1#2{
 	\draw[color=blue] \punkt{#1}{#2} -- \punkt{#1+1}{#2}
 		-- \punkt{#1}{(#2)+1} -- cycle;
+	\draw[->,color=blue] \punkt{#1}{#2} -- \punkt{#1+1}{#2};
 }
 
 \def\gebiet{
diff --git a/buch/chapters/95-homologie/images/homoclasses.pdf b/buch/chapters/95-homologie/images/homoclasses.pdf
index 217ae75..fbbaedd 100644
--- a/buch/chapters/95-homologie/images/homoclasses.pdf
+++ b/buch/chapters/95-homologie/images/homoclasses.pdf
diff --git a/buch/chapters/95-homologie/images/homoclasses.tex b/buch/chapters/95-homologie/images/homoclasses.tex
index e325d9b..4467f08 100644
--- a/buch/chapters/95-homologie/images/homoclasses.tex
+++ b/buch/chapters/95-homologie/images/homoclasses.tex
@@ -1,5 +1,5 @@
 %
-% tikztemplate.tex -- template for standalon tikz images
+% homoclasses.tex -- template for standalon tikz images
 %
 % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 %
@@ -11,11 +11,12 @@
 \usepackage{csvsimple}
 \usetikzlibrary{arrows,intersections,math}
 \begin{document}
-\def\skala{1}
+\def\skala{1.4}
 \begin{tikzpicture}[>=latex,thick,scale=\skala]
 
 \definecolor{darkgreen}{rgb}{0,0.6,0}
 \def\s{0.4}
+\def\h{-0.3}
 
 \def\punkt#1#2{({((#1)+0.5*(#2))*\s},{(#2)*\s*sqrt(3)/2})}
 \def\A{\punkt{0}{0}}
@@ -77,25 +78,29 @@
 \begin{scope}
 \gebiet
 \draw[color=darkgreen] \B -- \G -- \J -- \F -- cycle;
-\node[color=darkgreen] at ({2*\s},-0.5) {$z_5'$};
+\draw[->,color=darkgreen] \B -- \G;
+\node[color=darkgreen] at ({2*\s},{\h}) {$z_5'$};
 \end{scope}
 
 \begin{scope}[xshift=2cm]
 \gebiet
 \draw[color=darkgreen] \D -- \I -- \L -- \H -- cycle;
-\node[color=darkgreen] at ({2*\s},-0.5) {$z_6'$};
+\draw[->,color=darkgreen] \D -- \I;
+\node[color=darkgreen] at ({2*\s},{\h}) {$z_6'$};
 \end{scope}
 
 \begin{scope}[xshift=4cm]
 \gebiet
 \draw[color=darkgreen] \C -- \L -- \N -- \K -- \M -- \J -- cycle;
-\node[color=darkgreen] at ({2*\s},-0.5) {$z_9'$};
+\draw[->,color=darkgreen] \C -- \L;
+\node[color=darkgreen] at ({2*\s},{\h}) {$z_9'$};
 \end{scope}
 
 \begin{scope}[xshift=6cm]
 \gebiet
 \draw[color=darkgreen] \K -- \N -- \O -- \M -- cycle;
-\node[color=darkgreen] at ({2*\s},-0.5) {$z_{12}'$};
+\draw[->,color=darkgreen] \K -- \N;
+\node[color=darkgreen] at ({2*\s},{\h}) {$z_{12}'$};
 \end{scope}
 
 
diff --git a/buch/chapters/95-homologie/images/homocycles.pdf b/buch/chapters/95-homologie/images/homocycles.pdf
index 075bb65..b68519e 100644
--- a/buch/chapters/95-homologie/images/homocycles.pdf
+++ b/buch/chapters/95-homologie/images/homocycles.pdf
diff --git a/buch/chapters/95-homologie/images/homocycles.tex b/buch/chapters/95-homologie/images/homocycles.tex
index 898cac6..8f20a0c 100644
--- a/buch/chapters/95-homologie/images/homocycles.tex
+++ b/buch/chapters/95-homologie/images/homocycles.tex
@@ -75,79 +75,92 @@
 
 \begin{scope}
 \gebiet
-\draw[color=red] \A -- \B -- \F -- cycle;
+\draw[->,color=red] \A -- \B -- \F -- cycle;
+\draw[->,color=red] \A -- \B;
 \node[color=red] at ({2*\s},-0.5) {$z_1$};
 \end{scope}
 
 \begin{scope}[xshift=2cm]
 \gebiet
 \draw[color=red] \B -- \C -- \G -- cycle;
+\draw[->,color=red] \B -- \C;
 \node[color=red] at ({2*\s},-0.5) {$z_2$};
 \end{scope}
 
 \begin{scope}[xshift=4cm]
 \gebiet
 \draw[color=red] \C -- \D -- \H -- cycle;
+\draw[->,color=red] \C -- \D;
 \node[color=red] at ({2*\s},-0.5) {$z_3$};
 \end{scope}
 
 \begin{scope}[xshift=6cm]
 \gebiet
 \draw[color=red] \D -- \E -- \I -- cycle;
+\draw[->,color=red] \D -- \E;
 \node[color=red] at ({2*\s},-0.5) {$z_4$};
 \end{scope}
 
 \begin{scope}[xshift=8cm]
 \gebiet
 \draw[color=red] \A -- \B -- \G -- \F -- cycle;
+\draw[<-,color=red] \A -- \B;
 \node[color=red] at ({2*\s},-0.5) {$z_5$};
 \end{scope}
 
 \begin{scope}[xshift=10cm]
 \gebiet
 \draw[color=red] \C -- \D -- \I -- \H -- cycle;
+\draw[<-,color=red] \C -- \D;
 \node[color=red] at ({2*\s},-0.5) {$z_6$};
 \end{scope}
 
 \begin{scope}[xshift=12cm]
 \gebiet
 \draw[color=red] \A -- \B -- \G -- \J -- \F -- cycle;
+\draw[->,color=red] \A -- \B;
 \node[color=red] at ({2*\s},-0.5) {$z_7$};
 \end{scope}
 
 \begin{scope}[xshift=0cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \C -- \D -- \I -- \L -- \H -- cycle;
+\draw[->,color=red] \C -- \D;
 \node[color=red] at ({2*\s},-0.5) {$z_8$};
 \end{scope}
 
 \begin{scope}[xshift=2cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \A -- \B -- \C -- \H -- \L -- \K -- \J -- \F -- cycle;
+\draw[<-,color=red] \A -- \B;
 \node[color=red] at ({2*\s},-0.5) {$z_9$};
 \end{scope}
 
 \begin{scope}[xshift=4cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \J -- \K -- \M -- cycle;
+\draw[->,color=red] \J -- \K;
 \node[color=red] at ({2*\s},-0.5) {$z_{10}$};
 \end{scope}
 
 \begin{scope}[xshift=6cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \A -- \B -- \C -- \H -- \L -- \N -- \K -- \J -- \F -- cycle;
+\draw[->,color=red] \A -- \B;
 \node[color=red] at ({2*\s},-0.5) {$z_{11}$};
 \end{scope}
 
 \begin{scope}[xshift=8cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \J -- \K -- \N -- \M -- cycle;
+\draw[<-,color=red] \J -- \K;
 \node[color=red] at ({2*\s},-0.5) {$z_{12}$};
 \end{scope}
 
 \begin{scope}[xshift=10cm,yshift=-3cm]
 \gebiet
 \draw[color=red] \J -- \K -- \N -- \O -- \M -- cycle;
+\draw[->,color=red] \J -- \K;
 \node[color=red] at ({2*\s},-0.5) {$z_{13}$};
 \end{scope}