Add latex documentation

doc/latex/coordinate_transform.tex

@@ -0,0 +1,37 @@
+\begin{tikzpicture}
+\begin{axis}[grid=both,
+             ymin=0,
+             ymax=4.5,
+             xmax=4.5,
+             xmin=0,
+             xticklabel=\empty,
+             yticklabel=\empty,
+             minor tick num=1,
+             axis lines = middle,
+             xlabel=$x_1$,
+             ylabel=$x_2$,
+             label style = {at={(ticklabel cs:1.1)}}, 
+             axis equal=true, width=6cm, height=6cm]
+
+  \coordinate (t1ll) at (1, 1);
+  \pic at (t1ll) {ig_tree={t1, $\Omega_1$}};
+  \pic at (t1ll) {ig_tree_region={{0.8, 0.8}, {1.0, 1.0}, green}};
+
+  \coordinate (t2ll) at (3, 1);
+  \pic at (t2ll) {ig_tree={t2, $\Omega_2$}};
+  \pic at (t2ll) {ig_tree_region={{-0.2, 0.8}, {0.0, 1.0}, green}};
+
+  \coordinate (t3ll) at (3, 3);
+  \pic at (t3ll) {ig_tree={t3, $\Omega_3$}};
+  \pic at (t3ll) {ig_tree_region={{-0.2, -0.2}, {0.0, 0.0}, green}};
+
+  \path[thick, ->] ([shift={(0.9, 0.85)}]t1ll)  edge[bend right] node [below] 
+                   {$\tau_{1 \rightarrow 2}$} ([shift={(-0.15, 0.85)}]t2ll);
+
+  \path[thick, ->] ([shift={(-0.05, 0.95)}]t2ll)  edge[bend right] node [right] 
+                   {$\tau_{2 \rightarrow 3}$} ([shift={(-0.05, -0.15)}]t3ll);
+
+  \path[thick, ->] ([shift={(0.9, 0.95)}]t1ll)  edge node [left] 
+                   {$\tau_{1 \rightarrow 3}$} ([shift={(-0.15, -0.05)}]t3ll);
+\end{axis}
+\end{tikzpicture}

doc/latex/indexed_cube.tex

@@ -0,0 +1,28 @@
+\begin{tikzpicture}
+\newcommand{\Depth}{2}
+\newcommand{\Height}{2}
+\newcommand{\Width}{2}
+\coordinate (O) at (0,0,0);
+\coordinate (A) at (0,\Width,0);
+\coordinate (B) at (0,\Width,\Height);
+\coordinate (C) at (0,0,\Height);
+\coordinate (D) at (\Depth,0,0);
+\coordinate (E) at (\Depth,\Width,0);
+\coordinate (F) at (\Depth,\Width,\Height);
+\coordinate (G) at (\Depth,0,\Height);
+
+\draw[black] (O) node [left]{2} -- (C) node [left]{0} -- (G) node [right]{1} -- (D) node [right]{3} -- cycle;% Bottom Face
+\draw[black] (O) -- (A) -- (E) -- (D) -- cycle;% Back Face
+\draw[black] (O) -- (A) -- (B) -- (C) -- cycle;% Left Face
+\draw[black] (D) -- (E) -- (F) -- (G) -- cycle;% Right Face
+\draw[black] (C) -- (B) -- (F) -- (G) -- cycle;% Front Face
+\draw[black] (A) node [left]{6} -- (B) node [left]{4} -- (F) node [right]{5}-- (E) node [right]{7} -- cycle;% Top Face
+
+\path[->] ([shift={(0,0,0)}]C) edge node [below] {$x_1$} ([shift={(-1.0, 0, 0)}]G);
+\path[->] ([shift={(0,0,0)}]C) edge node [right] {$x_2$} ([shift={(0, 0, 1.0)}]O);
+\path[->] ([shift={(0,0,0)}]C) edge node [left] {$x_3$} ([shift={(0, -1.0, 0)}]B);
+%\draw[blue, fill=black] (1.0, 2.0, 1.0) circle (0.05) node[above] {$(0, 0, 1)$};
+%\draw[blue, fill=black] (1.0, 0.0, 1.0) circle (0.05) node[below] {$(0, 0, -1)$};
+%\draw[blue, fill=black] (2.0, 1.0, 1.0) circle (0.05) node[right] {$(1, 0, 0)$};
+%\draw[blue, fill=black] (0.0, 1.0, 1.0) circle (0.05) node[left] {$(-1, 0, 0)$};
+\end{tikzpicture}

doc/latex/mesh_overview.tex

@@ -0,0 +1,68 @@
+\begin{tikzpicture}[line/.style={red, very thick}]
+\coordinate (m2) (10.0, 0.0);
+
+\coordinate (n1) at (0,0); 
+\coordinate (n2) at (5,0);
+\coordinate (n3) at (2.5,4);
+\coordinate (n4) at (2.5,1.5);
+
+\fill[fill=red!5] (n1) to[out=-10,in=190] (n2) to[out=140,in=-35] (n3) to[out=-135,in=80] cycle;
+\shade[ball color=blue!10!white,opacity=0.20] (n1) to[out=-10,in=190] (n2) to[out=140,in=-35] (n3) to[out=-135,in=80] cycle;
+
+\draw[line] (n1) to[out=-10,in=190] coordinate[pos=.45] (m12) (n2);
+\draw[line] (n2) to[out=140,in=-35] coordinate[pos=.6] (m23) (n3);
+\draw[line] (n3) to[out=-135,in=80] coordinate[pos=.5] (m31) (n1);
+
+\draw[line] (n4) to[out=-85,in=85] (m12);
+\draw[line] (n4) to[out=45,in=225] (m23);
+\draw[line] (n4) to[out=140,in=-45] (m31);
+
+\node (m1) at ($0.25*(n1) + 0.25*(m12) + 0.25*(n4) + 0.25*(m31)$) {$\mathcal{M}_1$};
+\node (m2) at ($0.25*(m12) + 0.25*(n2) + 0.25*(m23) + 0.25*(n4)$) {$\mathcal{M}_2$};
+\node (m3) at ($0.25*(n4) + 0.25*(m23) + 0.25*(n3) + 0.25*(m31)$) {$\mathcal{M}_3$};
+
+\begin{axis}[grid=both,
+             ymin=0,
+             ymax=5.5,
+             xmax=5.5,
+             xmin=0,
+             xticklabel=\empty,
+             yticklabel=\empty,
+             minor tick num=1,
+             axis lines = middle,
+             xlabel=$x_1$,
+             ylabel=$x_2$,
+             label style = {at={(ticklabel cs:1.1)}}, 
+             axis equal=true, 
+             at={(6cm, 0)},
+             width=6.0cm, height=6.0cm]
+
+  \coordinate (t1ll) at (1, 1);
+  \pic at (t1ll) {ig_tree={t1, $\Omega_1$}};
+
+  \coordinate (t2ll) at (4, 1);
+  \pic at (t2ll) {ig_tree={t2, $\Omega_2$}};
+
+  \coordinate (t3ll) at (1, 4);
+  \pic at (t3ll) {ig_tree={t3, $\Omega_3$}};
+
+  \path[thick, ->, black!40] ([shift={(0.9,0.4)}]t1ll) edge[bend right] 
+          node [above] {$\tau_{1 \rightarrow 2}$} ([shift={(-0.1,0.4)}]t2ll);
+  \path[thick, ->, black!40] ([shift={(0.1,0.6)}]t2ll) edge[bend right] 
+          node [above] {$\tau_{2 \rightarrow 1}$} ([shift={(1.1,0.6)}]t1ll);
+
+  \path[thick, ->, black!40] ([shift={(0.6,0.9)}]t2ll) edge[bend right] 
+          node [right] {$\tau_{2 \rightarrow 3}$} ([shift={(1.1,0.6)}]t3ll);
+  \path[thick, ->, black!40] ([shift={(0.9,0.4)}]t3ll) edge[bend right] 
+          node [right] {$\tau_{3 \rightarrow 2}$} ([shift={(0.4, 1.1)}]t2ll);
+
+  \path[thick, ->, black!40] ([shift={(0.4,0.1)}]t3ll) edge[bend right] 
+          node [] {$\tau_{3 \rightarrow 1}$} ([shift={(0.4, 1.1)}]t1ll);
+  \path[thick, ->, black!40] ([shift={(0.6,0.9)}]t1ll) edge[bend right] 
+          node [] {$\tau_{1 \rightarrow 3}$} ([shift={(0.6, -0.1)}]t3ll);        
+\end{axis}
+
+\path[thick, ->] (m1.south) edge[bend right = 60] node [below] {$g_1^{-1}$} (t1.south);
+\path[thick, ->] (m2.south) edge[bend right = 50] node [below] {$g_2^{-1}$} (t2.south);
+\path[thick, ->] (m3.north) edge[bend left] node [above] {$g_3^{-1}$} (t3.west);
+\end{tikzpicture}

doc/latex/node_valence.tex

@@ -0,0 +1,51 @@
+\begin{tikzpicture}
+\coordinate (v3) at (3, 0); 
+\draw[thick] (v3) -- ([shift={(0, 1)}]v3) -- ([shift={(1, 1)}]v3) -- ([shift={(1, 0)}]v3) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v3) -- ([shift={(1, 1)}]v3) -- ([shift={(2, 2)}]v3) -- ([shift={(0, 2)}]v3) -- cycle;
+\draw[thick] ([shift={(1, 0)}]v3) -- ([shift={(2, 0)}]v3) -- ([shift={(2, 2)}]v3) -- ([shift={(1, 1)}]v3) -- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v3) circle (0.05);
+\draw[red] ([shift={(1, 1)}]v3) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_3$};
+
+\coordinate (v4) at (6, 0); 
+\draw[thick] (v4) -- ([shift={(0, 1)}]v4) -- ([shift={(1, 1)}]v4) -- ([shift={(1, 0)}]v4) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v4) -- ([shift={(1, 1)}]v4) -- ([shift={(1, 2)}]v4) -- ([shift={(0, 2)}]v4) -- cycle;
+\draw[thick] ([shift={(1, 0)}]v4) -- ([shift={(2, 0)}]v4) -- ([shift={(2, 1)}]v4) -- ([shift={(1, 1)}]v4) -- cycle;
+\draw[thick] ([shift={(1, 1)}]v4) -- ([shift={(2, 1)}]v4) -- ([shift={(2, 2)}]v4) -- ([shift={(1, 2)}]v4) -- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v4) circle (0.05);
+\draw[black] ([shift={(1, 1)}]v4) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_4$};
+
+\coordinate (v5) at (9, 0); 
+\draw[thick] (v5) -- ([shift={(0, 1)}]v5) -- ([shift={(1, 1)}]v5) -- ([shift={(1, 0)}]v5) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v5) -- ([shift={(1, 1)}]v5) -- ([shift={(1, 2)}]v5) -- ([shift={(0, 2)}]v5) -- cycle;
+\draw[thick] ([shift={(1, 0)}]v5) -- ([shift={(2, 0)}]v5) -- ([shift={(2, 1)}]v5) -- ([shift={(1, 1)}]v5) -- cycle;
+\draw[thick] ([shift={(1, 1)}]v5) -- ([shift={(2, 1)}]v5) -- ([shift={(2.5, 1.5)}]v5) -- ([shift={(1.75, 1.75)}]v5) -- cycle;
+\draw[thick] ([shift={(1, 1)}]v5) -- ([shift={(1, 2)}]v5) -- ([shift={(1.5, 2.5)}]v5) -- ([shift={(1.75, 1.75)}]v5) -- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v5) circle (0.05);
+\draw[red] ([shift={(1, 1)}]v5) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_5$};
+
+\coordinate (v1edge) at (0, -3); 
+\draw[thick] (v1edge) -- ([shift={(0, 1)}]v1edge) [red]-- ([shift={(1, 1)}]v1edge) [red]-- ([shift={(1, 0)}]v1edge) [black]-- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v1edge) circle (0.05);
+\draw[black] ([shift={(1, 1)}]v1edge) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_{2b}$};
+
+\coordinate (v2edge) at (3, -3); 
+\draw[thick] (v2edge) -- ([shift={(0, 1)}]v2edge) [red]-- ([shift={(1, 1)}]v2edge) [black]-- ([shift={(1, 0)}]v2edge) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v2edge) -- ([shift={(1, 1)}]v2edge) [red]-- ([shift={(1, 2)}]v2edge) [black]-- ([shift={(0, 2)}]v2edge) -- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v2edge) circle (0.05);
+\draw[black] ([shift={(1, 1)}]v2edge) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_{3b}$};
+
+\coordinate (v3edge) at (6, -3); 
+\draw[thick] (v3edge) -- ([shift={(0, 1)}]v3edge) [red]-- ([shift={(1, 1)}]v3edge) [black]-- ([shift={(1, 0)}]v3edge) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v3edge) -- ([shift={(1, 1)}]v3edge) [red]-- ([shift={(1, 2)}]v3edge) [black]-- ([shift={(0, 2)}]v3edge) -- cycle;
+\draw[thick] ([shift={(1, 0)}]v3edge) -- ([shift={(2, 0)}]v3edge) -- ([shift={(2, 1)}]v3edge) [red]-- ([shift={(1, 1)}]v3edge) [black]-- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v3edge) circle (0.05);
+\draw[black] ([shift={(1, 1)}]v3edge) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_{4b}$};
+
+\coordinate (v4edge) at (9, -3); 
+\draw[thick] (v4edge) -- ([shift={(0, 1)}]v4edge) -- ([shift={(1, 1)}]v4edge) -- ([shift={(1, 0)}]v4edge) -- cycle;
+\draw[thick] ([shift={(0, 1)}]v4edge) -- ([shift={(1, 1)}]v4edge) [red]-- ([shift={(0.9, 2)}]v4edge) [black]-- ([shift={(0, 2)}]v4edge) -- cycle;
+\draw[thick] ([shift={(1, 0)}]v4edge) -- ([shift={(2, 0)}]v4edge) -- ([shift={(2, 1)}]v4edge) -- ([shift={(1, 1)}]v4edge) -- cycle;
+\draw[thick] ([shift={(1, 1)}]v4edge) -- ([shift={(2, 1)}]v4edge) -- ([shift={(2, 2)}]v4edge) -- ([shift={(1.1, 2)}]v4edge) [red]-- cycle;
+\draw[blue, fill=blue] ([shift={(1, 1)}]v4edge) circle (0.05);
+\draw[black] ([shift={(1, 1)}]v4edge) circle (0.2) node[shift={(-0.25, -0.25)}] {$n_{5b}$};
+\end{tikzpicture}

doc/latex/offset_cube.tex

@@ -0,0 +1,41 @@
+\begin{tikzpicture}
+\newcommand{\Depth}{2}
+\newcommand{\Height}{2}
+\newcommand{\Width}{2}
+\coordinate (O) at (0,0,0);
+\coordinate (A) at (0,\Width,0);
+\coordinate (B) at (0,\Width,\Height);
+\coordinate (C) at (0,0,\Height);
+\coordinate (D) at (\Depth,0,0);
+\coordinate (E) at (\Depth,\Width,0);
+\coordinate (F) at (\Depth,\Width,\Height);
+\coordinate (G) at (\Depth,0,\Height);
+
+\draw[black] (O) node [left]{} -- (C) node [left]{} -- (G) node [right]{} -- (D) node [right]{} -- cycle;% Bottom Face
+\draw[black] (O) -- (A) -- (E) -- (D) -- cycle;% Back Face
+\draw[black] (O) -- (A) -- (B) -- (C) -- cycle;% Left Face
+\draw[black] (D) -- (E) -- (F) -- (G) -- cycle;% Right Face
+\draw[black] (C) -- (B) -- (F) -- (G) -- cycle;% Front Face
+\draw[black] (A) node [left]{} -- (B) node [left]{} -- (F) node [right]{}-- (E) node [right]{} -- cycle;% Top Face
+
+\draw[blue, fill=blue] (1.0, 2.0, 1.0) circle (0.05) node[above] {$(0, 0, 1)$};
+\draw[blue, fill=blue] (1.0, 0.0, 1.0) circle (0.05) node[below] {$(0, 0, -1)$};
+\draw[blue, fill=blue] (2.0, 1.0, 1.0) circle (0.05) node[right] {$(1, 0, 0)$};
+\draw[blue, fill=blue] (0.0, 1.0, 1.0) circle (0.05) node[left] {$(-1, 0, 0)$};
+%\draw[blue, fill=black] (1.0, 1.0, 2.0) circle (0.05) node[right] {$(0, 1, 0)$};
+%\draw[blue, fill=black] (1.0, 1.0, 0.0) circle (0.05) node[left] {$(0, -1, 0)$};
+
+\draw[green, fill=green] (1.0, 2.0, 2.0) circle (0.05) node[above] {$(0, -1, 1)$};
+\draw[green, fill=green] (1.0, 0.0, 2.0) circle (0.05) node[below] {$(0, -1, -1)$};
+\draw[green, fill=green] (2.0, 1.0, 2.0) circle (0.05) node[right] {$(1, -1, 0)$};
+\draw[green, fill=green] (0.0, 1.0, 2.0) circle (0.05) node[left] {$(-1, -1, 0)$};
+
+\draw[red, fill=red] (2.0, 2.0, 0.0) circle (0.05) node[above] {$(1, 1, 1)$};
+\draw[red, fill=red] (2.0, 0.0, 0.0) circle (0.05) node[below] {$(1, 1, -1)$};
+\draw[red, fill=red] (0.0, 2.0, 0.0) circle (0.05) node[left] {$(-1, 1, 1)$};
+\draw[red, fill=red] (0.0, 0.0, 0.0) circle (0.05) node[left] {$(-1, 1, -1)$};
+
+\path[->] ([shift={(0,0,0)}]C) edge node [below] {$x_1$} ([shift={(-1.0, 0, 0)}]G);
+\path[->] ([shift={(0,0,0)}]C) edge node [right] {$x_2$} ([shift={(0, 0, 1.0)}]O);
+\path[->] ([shift={(0,0,0)}]C) edge node [left] {$x_3$} ([shift={(0, -1.0, 0)}]B);
+\end{tikzpicture}