blob: comparison text/appendixes/comparing

equal deleted inserted replaced

-:bbf14d934cb1
+:27c075c68816
 \foreach \y in {1,0.86,...,-1} {
 	\draw[green!50!brown] (0,\y)--(1,\y);
 }
 \end{scope}
 \draw[->,blue] (1.5,-1) node[below] {$a \times I$} -- (0.75,0);
-\end{tikzpicture} \\
+\end{tikzpicture} &
 \begin{tikzpicture}[baseline]
 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
 \draw[dashed] (c.north) -- (c.south);
 \node[right=6] at (c) {$g$};
 \node[left=6] at (c) {$f$};
 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
 \node[draw,dashed] (d) at (0,0) [circle, minimum height=1cm,minimum width=1cm] {};
 \draw[dashed] (c.north) -- (d.north) (d.south) -- (c.south);
 \node[right=18] at (c) {$g$};
 \node[left=18] at (c) {$f$};
+\draw[->,blue] (.75,-1.25) node[below] {$a \times I$} -- (0,-0.25);
 \clip (0,0) circle (0.5cm);
 \foreach \y in {1,0.86,...,-1} {
 	\draw[green!50!brown] (-1,\y)--(1,\y);
 }
-\end{tikzpicture} \\
+\end{tikzpicture} &
 \begin{tikzpicture}[baseline]
 \begin{scope}
 \clip (-1.3,-2) rectangle (0,2);
 \node[draw] (c) at (0,0) [ellipse, minimum height=2cm,minimum width=2.5cm] {};
 \end{scope}
 	}
 \end{scope}
 \draw (m.center) .. controls +(1,0) and +(1,0) .. (c.south);
 \node[right=15,below=8] at (c) {$g$};
 \node[left=12] at (c) {$f$};
+\draw[->,blue] (.65,-1.25) node[below] {$a \times I$} -- (0,-0.25);
 \end{tikzpicture}
 \end{align*}
 \caption{Examples of inserting or deleting product regions.}
 \label{fig:product-regions}
 \end{figure}
 \draw[blue,->] (-0.8,-1.2) node[below] {$(a \bullet d) \times I$} -- (1,-0.5) ;
 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
 .. controls +(1,.4) and +(-1,1) .. (g2)
 .. controls +(-1,.4) and +(1,.4) .. (g1)
 .. controls +(-1,-.4) and +(1,-1) .. (f1);
-\foreach \x in {0,0.1, ..., 6} {
+\foreach \x in {0,0.2, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 }
 \end{tikzpicture}
 };
 \node (fg2) at (4,-4) {
 \begin{scope}
 \path[clip] (f1) .. controls +(1,-.4) and +(-1,-.4) .. (f2)
 .. controls +(1,.4) and +(-1,1) .. (g2)
 .. controls +(-1,.4) and +(1,.4) .. (g1)
 .. controls +(-1,-.4) and +(1,-1) .. (f1);
-\foreach \x in {0,0.1, ..., 6} {
+\foreach \x in {0,0.2, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 }
 \end{scope}
 \begin{scope}
 \path[clip] (f1) ..  controls +(1,1.5) and +(-1,2).. (g2)
 		      .. controls +(-1,1) and +(1,.4) .. (f2)
 		      .. controls +(-1,.8) and + (1,.8) .. (f1);
-\foreach \x in {0,0.1, ..., 6} {
+\foreach \x in {0,0.2, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 }
 \end{scope}
 \end{tikzpicture}
 };
 \draw[blue,->] (4,1.75) node[above] {$(b \bullet d) \times I$}-- + (0,-1);
 \begin{scope}
 \path[clip] (f1) ..  controls +(1,1.5) and +(-1,1.5).. (g2)
 		      .. controls +(-1,.8) and +(1,.8) .. (f2)
 		      .. controls +(-1,.8) and + (1,.8) .. (f1);
-\foreach \x in {0,0.1, ..., 6} {
+\foreach \x in {0,0.2, ..., 6} {
 	\draw[green!50!brown] (\x,-2) -- + (0,4);
 }
 \end{scope}
 \end{tikzpicture}
 };
 %\mathfig{0.4}{triangle/triangle4a} \\
 %\mathfig{0.4}{triangle/triangle4b} \\
 %\mathfig{0.4}{triangle/triangle4c}
 %\end{align*}
 \begin{align*}
+u & =
+\begin{tikzpicture}[baseline]
+\coordinate (P) at (0,0);
+\coordinate (Q) at (3,0);
+\coordinate (R) at (1.5,-0.5);
+\draw (P) \vertex to[out=40,in=140] node[above] {$a$} (Q) \vertex -- node[below] {\scriptsize $y \times I$} (R) \vertex to[out=-180,in=-40] node[below] {$a$} (P);
+\clip (P) to[out=40,in=140] (Q) -- (R) to[out=-180,in=-40] (P);
+\foreach \x in {1,...,9} {
+	\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q);
+	\path (P) to[out=-40,in=-180] node[pos=\x/10] (PR\x) {} (R);
+	\draw[green!50!brown] (PQ\x.center) -- (PR\x.center);
+}
+\end{tikzpicture} &
+v & =
+\begin{tikzpicture}[baseline]
+\coordinate (P) at (0,0);
+\coordinate (Q) at (3,0);
+\coordinate (R) at (1.5,-0.5);
+\draw (P) \vertex to[out=40,in=140] node[above] {$b$} (Q) \vertex to[out=-140,in=0] node[below] {$b$} (R) -- node[below] {\scriptsize $y \times I$}  (P);
+\clip (P) to[out=40,in=140] (Q) to[out=-140,in=0] (R) -- (P);
+\foreach \x in {1,...,9} {
+	\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q);
+	\path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q);
+	\draw[brown] (PQ\x.center) -- (RQ\x.center);
+}
+\end{tikzpicture}
+\end{align*}
+\vspace{-2cm}
+\begin{align*}
 \alpha & =
 \begin{tikzpicture}[baseline]
 \node[circle,fill=black,inner sep=1pt] at (1.73,0) {};
 \node[circle,fill=black,inner sep=1pt] at (-1.73,0) {};
 \begin{scope}[yshift=-1cm]
 		}
 		\foreach \t in {-1.1,-0.9,...,0} {
 			\draw[blue!50!brown] (\t,-1) -- +(1.19,2);
 		}
 		\foreach \t in {0.1,0.3,...,2.5} {
-			\draw[red!50!brown] (\t,-1) -- +(1.19,2);
+			\draw[brown] (\t,-1) -- +(1.19,2);
 		}
 	\foreach \x in {0,1,...,10} {
 		\path (L1) to[out=60,in=-120] node[pos=\x/10] (p1\x) {}(U1);
 		\path (L2) to[out=60,in=-120] node[pos=\x/10] (p2\x) {} (U2);
 		\draw[blue!50!brown] (p1\x.center) -- (p2\x.center);
 	}
 \end{scope}
 \draw[->, thick, blue!75!yellow] (-2,0.5) node[black, left] {$a\times I$} -- (-1,0.5);
 \draw[->, thick, blue!75!yellow] (-0.1,-1.5) node[black, below] {$y\times I \times I$} -- (-0.25,-0.5);
 \draw[->, thick, blue!75!yellow] (2,-0.5) node[black, right] {$b\times I$} -- (1,-0.5);
-\end{tikzpicture} \\
+\end{tikzpicture}
-u & =
-\begin{tikzpicture}[baseline]
-\coordinate (P) at (0,0);
-\coordinate (Q) at (3,0);
-\coordinate (R) at (1.5,-0.5);
-\draw (P) \vertex to[out=40,in=140] node[above] {$a$} (Q) \vertex -- node[below] {\scriptsize $y \times I$} (R) \vertex to[out=-180,in=-40] node[below] {$a$} (P);
-\clip (P) to[out=40,in=140] (Q) -- (R) to[out=-180,in=-40] (P);
-\foreach \x in {1,...,9} {
-	\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q);
-	\path (P) to[out=-40,in=-180] node[pos=\x/10] (PR\x) {} (R);
-	\draw[green!50!brown] (PQ\x.center) -- (PR\x.center);
-}
-\end{tikzpicture} \\
-v & =
-\begin{tikzpicture}[baseline]
-\coordinate (P) at (0,0);
-\coordinate (Q) at (3,0);
-\coordinate (R) at (1.5,-0.5);
-\draw (P) \vertex to[out=40,in=140] node[above] {$b$} (Q) \vertex to[out=-140,in=0] node[below] {$b$} (R) -- node[below] {\scriptsize $y \times I$}  (P);
-\clip (P) to[out=40,in=140] (Q) to[out=-140,in=0] (R) -- (P);
-\foreach \x in {1,...,9} {
-	\path (P) to[out=40,in=140] node[pos=\x/10] (PQ\x) {} (Q);
-	\path (R) to[out=0,in=-140] node[pos=\x/10] (RQ\x) {} (Q);
-	\draw[brown] (PQ\x.center) -- (RQ\x.center);
-}
-\end{tikzpicture} \\
 \end{align*}
+\vspace{-1cm}
 \caption{Ingredients for the triangle axiom.}
 \label{fig:ingredients-triangle-axiom}
 \end{figure}
 The horizontal compositions $u *_h \id_b$ and $\id_a *_h  v$ are shown in Figure \ref{fig:horizontal-composition}
 \foreach \n in {1,...,7} {
 	\path (M) to[out=-20,in=-150] node[coordinate,pos=\n/8] (MR\n) {} (R);
 	\path (T) to[out=0,in=140] node[coordinate,pos=\n/8] (TR\n) {} (R);
 	\draw[brown] (MR\n) -- (TR\n);
 }
-\end{tikzpicture} \\
+\end{tikzpicture} &
 (a \times I) *_h v & =
 \begin{tikzpicture}[baseline]
 \coordinate (L) at (0,0);
 \coordinate (R) at (3,0);
 \coordinate (T) at (1.5,0.8);
 \end{align*}
 \caption{Horizontal compositions in the triangle axiom.}
 \label{fig:horizontal-composition}
 \end{figure}
 \begin{figure}[t]
+\vspace{-1.5cm}
 \begin{align*}
 %\mathfig{0.4}{triangle/triangle4f} \\
 \begin{tikzpicture}
 \node[circle,fill=black,inner sep=1pt] (A) at (1.73,0) {};
 \node[circle,fill=black,inner sep=1pt] (B) at (-1.73,0) {};
 	\draw[green!50!brown] (BC\n) -- +(-1.5,-1);
 }
 \foreach \n in {0,...,5} {
 	\path (C) to[out=0,in=-180] node[coordinate,pos=\n/5] (CD\n) {} (D);
 	\path (L1) to[out=-30,in=-180] node[coordinate,pos=\n/5] (L1L2\n) {} (L2);
-	\draw[blue] (CD\n) -- (L1L2\n);
+	\draw[blue!50!brown] (CD\n) -- (L1L2\n);
 }
 \foreach \n in {0,...,5} {
 	\path (L1) to[out=40,in=-140] node[coordinate,pos=\n/5] (L1C\n) {} (C);
 	\path (L2) to[out=50,in=-130] node[coordinate,pos=\n/5] (L2D\n) {} (D);
-	\draw[blue] (L1C\n) -- (L2D\n);
+	\draw[blue!50!brown] (L1C\n) -- (L2D\n);
 }
 \foreach \n in {1,3,...,9} {
 	\path(C) to[out=30,in=150] node[coordinate,pos=\n/10] (CA\n) {} (A);
 	\path(D) to[out=0,in=180] node[coordinate,pos=\n/10] (DA\n) {} (A);
 	\path(L2) to[out=0,in=-120] node[coordinate,pos=\n/10] (L2A\n) {} (A);
 }
 \draw[dashed] (U) -- (C) -- (L1) (L2) -- (D);
 \end{scope}
 \end{tikzpicture}
 \end{align*}
+\vspace{-2.5cm}
 \caption{Vertical composition in the triangle axiom.}
 \label{fig:vertical-composition}
 \end{figure}
 \begin{figure}[t]
 %\begin{align*}
 }
 \foreach \n in {1,...,5} {
 	\path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X);
 	\path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X);
 	\path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M');
-	\draw[blue] (B'X\n) -- (M'X\n);
+	\draw[blue!50!brown] (B'X\n) -- (M'X\n);
-	\draw[blue] (B'M'\n) -- (B'X\n);
+	\draw[blue!50!brown] (B'M'\n) -- (B'X\n);
 }
 \end{tikzpicture}
 \xrightarrow{\text{collar}}
 \begin{tikzpicture}[baseline]
 \coordinate (L) at (0,0);
 }
 \foreach \n in {1,...,5} {
 	\path (B') to[out=-30,in=180] node[coordinate,pos=\n/6] (B'X\n) {} (X);
 	\path (M') to[out=-90,in=90] node[coordinate,pos=\n/6] (M'X\n) {} (X);
 	\path (B') to[out=30,in=-150] node[coordinate,pos=\n/6] (B'M'\n) {} (M');
-	\draw[blue] (B'X\n) -- (M'X\n);
+	\draw[blue!50!brown] (B'X\n) -- (M'X\n);
-	\draw[blue] (B'M'\n) -- (B'X\n);
+	\draw[blue!50!brown] (B'M'\n) -- (B'X\n);
 }
 \end{tikzpicture}
 \end{align*}
 \caption{Adding a collar in the proof of the triangle axiom.}
 \label{fig:adding-a-collar}

changeset 972	27c075c68816
parent 970	7f47bf84b0f1