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commit ac83fff884120b1d8f6e3f502bf68fe58be50f04
parent 3371aa4940d1a6bee2b8a0a85305ddae52622895
Author: Pablo <pablo-escobar@riseup.net>
Date:   Wed, 24 Feb 2021 21:02:43 +0000

Added more LaTeX macros, updated formating of the TikZ pictures and added a picture for the correspondance between conjugation by unitary quaternions and rotations in 3D space

Diffstat:
M.local/share/texmf/tex/latex/images/dihedral-representation-is-irreducible.tikz | 10+++++-----
M.local/share/texmf/tex/latex/images/dihedral-representation.tikz | 8++++----
M.local/share/texmf/tex/latex/images/euclidian-plain.tikz | 2+-
M.local/share/texmf/tex/latex/images/hiperbolic-plain.tikz | 4++--
M.local/share/texmf/tex/latex/images/projective-system-universal-property.tikz | 12++++++------
M.local/share/texmf/tex/latex/images/projective-system.tikz | 6+++---
A.local/share/texmf/tex/latex/images/quaternion-rotation.tikz | 43+++++++++++++++++++++++++++++++++++++++++++
M.local/share/texmf/tex/latex/images/smooth-function.tikz | 20++++++++++----------
M.local/share/texmf/tex/latex/images/smooth-manifold.tikz | 16++++++++--------
M.local/share/texmf/tex/latex/images/sphere-quotient.tikz | 6+++---
M.local/share/texmf/tex/latex/images/unit-circle-covering.tikz | 2+-
M.local/share/texmf/tex/latex/images/unit-circle.tikz | 4++--
M.local/share/texmf/tex/latex/images/velocity.tikz | 2+-
M.local/share/texmf/tex/latex/xalgebra.sty | 16+++++++++-------
14 files changed, 98 insertions(+), 53 deletions(-)
diff --git a/.local/share/texmf/tex/latex/images/dihedral-representation-is-irreducible.tikz b/.local/share/texmf/tex/latex/images/dihedral-representation-is-irreducible.tikz
@@ -2,19 +2,19 @@
 % irreducible
 \begin{tikzpicture}[scale=1.2]
     % The axis
-    \draw[->] (-1,0)--(3,0) node[right]{\(x\)};
-    \draw[->] (0,-0.5)--(0,1.5) node[above]{\(y\)};
+    \draw[->] (-1,0)--(3,0) node[right]{$x$};
+    \draw[->] (0,-0.5)--(0,1.5) node[above]{$y$};
     
     % sigma
-    \draw[->] (1.5,0.3) arc (0:50:1cm) node at (1.7,0.85) {\(\sigma\)};
+    \draw[->] (1.5,0.3) arc (0:50:1cm) node at (1.7,0.85) {$\sigma$};
 
     % A non-zero vector v
     \filldraw[black] (1.5, 0.3) circle (1pt);
-    \node[right] at (1.5, 0.3) {\(v\)};
+    \node[right] at (1.5, 0.3) {$v$};
 
     % sigma(v)
     \filldraw[black] (1, 1.15) circle (1pt);
-    \node[left] at (1, 1.15) {\(\sigma \cdot v\)};
+    \node[left] at (1, 1.15) {$\sigma \cdot v$};
     
     % The subspace spaned by v
     \draw[dotted] (-1, -0.2)--(3, 0.6);
diff --git a/.local/share/texmf/tex/latex/images/dihedral-representation.tikz b/.local/share/texmf/tex/latex/images/dihedral-representation.tikz
@@ -1,17 +1,17 @@
 % This picture represents the action of the dihedral group in the real plain
 \begin{tikzpicture}
   % The axis
-  \draw[->] (-3,0)--(3,0) node[right]{\(x\)};
-  \draw[->] (0,-2)--(0,2) node[above]{\(y\)};
+  \draw[->] (-3,0)--(3,0) node[right]{$x$};
+  \draw[->] (0,-2)--(0,2) node[above]{$y$};
    
   % The triangle
   \node[draw, regular polygon, regular polygon sides=3, minimum height=2cm]{};
   
   % The action of sigma
   \draw[->] (1.5, 0.5) arc (0:90:1cm) 
-            node at (1.5, 1.4) {\(\sigma\)};
+            node at (1.5, 1.4) {$\sigma$};
 
   % The action of tau
-  \draw[<->] (-1, -1)--(1, -1) node[right]{\(\tau\)};
+  \draw[<->] (-1, -1)--(1, -1) node[right]{$\tau$};
 \end{tikzpicture}   
 
diff --git a/.local/share/texmf/tex/latex/images/euclidian-plain.tikz b/.local/share/texmf/tex/latex/images/euclidian-plain.tikz
@@ -1,6 +1,6 @@
 % This picture represents the cartesian plain
 \begin{tikzpicture}
-    \draw (0, 0) node[left]{\(\mathbb{R}^2\)} -- 
+    \draw (0, 0) node[left]{$\mathbb{R}^2$} -- 
           (2, 0) -- 
           (3, 1) -- 
           (1, 1) -- cycle;
diff --git a/.local/share/texmf/tex/latex/images/hiperbolic-plain.tikz b/.local/share/texmf/tex/latex/images/hiperbolic-plain.tikz
@@ -1,8 +1,8 @@
 % This picture represents the hyperbolic plain
 \begin{tikzpicture}
     % The axis
-    \draw[dotted, ->] (0,  0) -- (6, 0) node[right]{\(x\)};
-    \draw[->] (1, -1) -- (1, 3) node[above]{\(y\)};
+    \draw[dotted, ->] (0,  0) -- (6, 0) node[right]{$x$};
+    \draw[->] (1, -1) -- (1, 3) node[above]{$y$};
     
     % A strait line (in the hyperbolic plain)
     \draw (2, 0) arc (180:0:1.5);
diff --git a/.local/share/texmf/tex/latex/images/projective-system-universal-property.tikz b/.local/share/texmf/tex/latex/images/projective-system-universal-property.tikz
@@ -11,22 +11,22 @@
     X_i \&                 \& X_j \\};
 
   % The morphism
-  \draw[->] (m-2-2) -- node[above right]{\(\pi_j\)} (m-3-3);
-  \draw[->] (m-2-2) -- node[above left]{\(\pi_i\)} (m-3-1);
+  \draw[->] (m-2-2) -- node[above right]{$\pi_j$} (m-3-3);
+  \draw[->] (m-2-2) -- node[above left]{$\pi_i$} (m-3-1);
 
   % The projections
-  \draw[dotted, ->] (m-1-2) -- node[right]{\(\theta\)} (m-2-2);
+  \draw[dotted, ->] (m-1-2) -- node[right]{$\theta$} (m-2-2);
 
   % The arrow from the projective system
-  \draw[->] (m-3-3) -- node[below]{\(\phi_{i, j}\)} (m-3-1);
+  \draw[->] (m-3-3) -- node[below]{$\phi_{i, j}$} (m-3-1);
 
   % The arrows from the compatible family of morphisms
   \draw[->] (m-1-2) 
             to[relative, out=-30, in=-150] 
-            node[left]{\(\theta_i\)}
+            node[left]{$\theta_i$}
             (m-3-1);
   \draw[->] (m-1-2) 
             to[relative, out=30, in=150] 
-            node[right]{\(\theta_j\)}
+            node[right]{$\theta_j$}
             (m-3-3);
 \end{tikzpicture}
diff --git a/.local/share/texmf/tex/latex/images/projective-system.tikz b/.local/share/texmf/tex/latex/images/projective-system.tikz
@@ -7,8 +7,8 @@
     X_i \&     \& X_k \\};
 
   % The arrows
-  \draw[->] (m-2-3) -- node[below]{\(\phi_{i, k}\)} (m-2-1);
-  \draw[->] (m-1-2) -- node[above left]{\(\phi_{i, j}\)} (m-2-1);
-  \draw[->] (m-2-3) -- node[above right]{\(\phi_{j, k}\)} (m-1-2);
+  \draw[->] (m-2-3) -- node[below]{$\phi_{i, k}$} (m-2-1);
+  \draw[->] (m-1-2) -- node[above left]{$\phi_{i, j}$} (m-2-1);
+  \draw[->] (m-2-3) -- node[above right]{$\phi_{j, k}$} (m-1-2);
 \end{tikzpicture}
 
diff --git a/.local/share/texmf/tex/latex/images/quaternion-rotation.tikz b/.local/share/texmf/tex/latex/images/quaternion-rotation.tikz
@@ -0,0 +1,43 @@
+% This drawing represents the correspondance between conjugation by unitary
+% quaternions and rotations in the 3-dimensional euclidian space
+\begin{tikzpicture}
+  % The rotation axis
+  \begin{scope}[rotate=-60]
+    % The axis
+    \draw[->] (0, 0) -- (0, 2.5) node[right]{$\mathrm{Im}\,q$};
+
+    % The rotation
+    \draw[->] (0.43, 2) node[right]{$\theta=\frac{\arccos(\mathrm{Re}\,q)}{2}$}
+                        arc (30:360:0.5 and 0.25);
+
+    % The origin and the intersection of ratation axis with the unit sphere
+    \filldraw (0, 0) circle (1pt) (0, 1) circle (1pt);
+  \end{scope}
+
+  % The circunference
+  \draw (0, 0) circle (1.5);
+
+  % The equator
+  \begin{scope}
+    \clip (-1.5, 0) rectangle (1.5, -1.5);
+    \draw ellipse (1.5 and 0.5);
+  \end{scope}
+
+  % The equator (on the other side of the sphere)
+  \begin{scope}
+    \clip (-1.5, 0) rectangle (1.5, 1.5);
+    \draw[dotted] ellipse (1.5 and 0.5);
+  \end{scope}
+
+  % Greenwhich
+  \begin{scope}
+    \clip (-1.5, -1.5) rectangle (0, 1.5);
+    \draw ellipse (0.5 and 1.5);
+  \end{scope}
+
+  % Greenwhich (on the other side of the sphere)
+  \begin{scope}
+    \clip (1.5, -1.5) rectangle (0, 1.5);
+    \draw[dotted] ellipse (0.5 and 1.5);
+  \end{scope}
+\end{tikzpicture}
diff --git a/.local/share/texmf/tex/latex/images/smooth-function.tikz b/.local/share/texmf/tex/latex/images/smooth-function.tikz
@@ -3,23 +3,23 @@
   % The manifolds
   \begin{scope}[shift={(-2, 0)}]
     \path pic {manifold};
-    \draw (0.5, -0.5) node[right]{\(M\)};
+    \draw (0.5, -0.5) node[right]{$M$};
   \end{scope}
   \begin{scope}[shift={(2, 0)}]
     \path pic {manifold};
-    \draw (0.5, -0.5) node[right]{\(N\)};
+    \draw (0.5, -0.5) node[right]{$N$};
   \end{scope}
 
   % The functions
-  \draw[->] (-1,  0.9) to[out=30, in=150] node[above]{\(f\)} (1, 0.9);
+  \draw[->] (-1,  0.9) to[out=30, in=150] node[above]{$f$} (1, 0.9);
   \draw[->] (-1, -1.7) to[out=30, in=150] 
-  node[above]{\(\psi \circ f \circ \varphi^{-1}\)} (1, -1.7);
-  \draw[->] (-2, -0.7) -- node[left]{\(\varphi\)} +(0, -1);
-  \draw[->] ( 2, -0.7) -- node[left]{\(\psi\)} +(0, -1);
+  node[above]{$\psi \circ f \circ \varphi^{-1}$} (1, -1.7);
+  \draw[->] (-2, -0.7) -- node[left]{$\varphi$} +(0, -1);
+  \draw[->] ( 2, -0.7) -- node[left]{$\psi$} +(0, -1);
 
   % The open sets in euclidian space
-  \draw[dash dot] (-2, -3) node{\(\varphi(U)\)} ellipse (1 and 0.5);
-  \draw[dash dot] ( 2, -3) node{\(\psi(V)\)} ellipse (1 and 0.5);
-  \draw (-3.5, -4) node[left]{\(\mathbb{R}^m\)} rectangle +(3, 2);
-  \draw ( 0.5, -4) node[left]{\(\mathbb{R}^n\)} rectangle +(3, 2);
+  \draw[dash dot] (-2, -3) node{$\varphi(U)$} ellipse (1 and 0.5);
+  \draw[dash dot] ( 2, -3) node{$\psi(V)$} ellipse (1 and 0.5);
+  \draw (-3.5, -4) node[left]{$\mathbb{R}^m$} rectangle +(3, 2);
+  \draw ( 0.5, -4) node[left]{$\mathbb{R}^n$} rectangle +(3, 2);
 \end{tikzpicture}
diff --git a/.local/share/texmf/tex/latex/images/smooth-manifold.tikz b/.local/share/texmf/tex/latex/images/smooth-manifold.tikz
@@ -4,14 +4,14 @@
   \begin{scope}[shift={(-0.5, 0)}]
     % The frontire
     \path pic {big-manifold};
-    \draw (-4, 2) node[below]{\(M\)};
+    \draw (-4, 2) node[below]{$M$};
 
     % The open sets of the manifold and their intersection
     \begin{scope}[shift={(1, 2.2)}]
       \draw (-0.5, 0) ellipse (0.7 and 0.5);
-      \draw (-1.2, 0) node[right]{\(U\)};
+      \draw (-1.2, 0) node[right]{$U$};
       \draw (0.5, 0) ellipse (0.7 and 0.5);
-      \draw (1.2, 0) node[left]{\(V\)};
+      \draw (1.2, 0) node[left]{$V$};
       \clip (-0.5, 0) ellipse (0.7 and 0.5);
       \draw[pattern=north west lines] (0.5, 0) ellipse (0.7 and 0.5);
     \end{scope}
@@ -19,16 +19,16 @@
 
   % The cards
   \draw[->] (-0.7, 1.7) to[relative, out=-20, in=-160] 
-            node[left]{\(\varphi_U\)} (-2, -0.7);
+            node[left]{$\varphi_U$} (-2, -0.7);
   \draw[->] (1.7, 1.7) to[relative, out=20, in=160] 
-            node[right]{\(\varphi_V\)} (2, -0.7);
+            node[right]{$\varphi_V$} (2, -0.7);
 
   % The corresponding open sets in euclidian space
   \begin{scope}[shift={(0, -1.5)}]
     % The open sets
     \draw (-2, 0) ellipse (1 and 0.6) (2, 0) ellipse (1 and 0.6);
-    \draw (-3, 0.5) node[above]{\(\varphi_U(U)\)};
-    \draw (3, 0.5) node[above]{\(\varphi_V(V)\)};
+    \draw (-3, 0.5) node[above]{$\varphi_U(U)$};
+    \draw (3, 0.5) node[above]{$\varphi_V(V)$};
 
     % The intersections
     \begin{scope}
@@ -42,7 +42,7 @@
 
     % The diffeomorphism
     \draw[->] (-0.9, 0) -- 
-              node[above]{\(\varphi_V \circ \varphi_U^{-1}\)} (0.9, 0);
+              node[above]{$\varphi_V \circ \varphi_U^{-1}$} (0.9, 0);
   \end{scope}
 \end{tikzpicture}
 
diff --git a/.local/share/texmf/tex/latex/images/sphere-quotient.tikz b/.local/share/texmf/tex/latex/images/sphere-quotient.tikz
@@ -31,9 +31,9 @@
     
     \draw (-3, 2.625) -- (1.5, 2.625) 
                       -- (3, 3.375) 
-                      node[right]{\(T_p S^n\)}
+                      node[right]{$T_p S^n$}
                       -- (-1.5, 3.375) 
                       -- cycle;
-    \filldraw[black] (0, 3) circle (2pt) node[right]{\(p\)};
-    \draw (0, 3.375) node[above]{\(\SO_n(\RR)\)};
+    \filldraw[black] (0, 3) circle (2pt) node[right]{$p$};
+    \draw (0, 3.375) node[above]{$\SO_n(\RR)$};
 \end{tikzpicture}
diff --git a/.local/share/texmf/tex/latex/images/unit-circle-covering.tikz b/.local/share/texmf/tex/latex/images/unit-circle-covering.tikz
@@ -11,6 +11,6 @@
     \end{axis}
 \end{scope}
 
-\draw[thick, ->] (0, 0) -- node[right]{\(p\)} (0, -1);
+\draw[thick, ->] (0, 0) -- node[right]{$p$} (0, -1);
 \draw[thick] (0, -1.5) ellipse (0.9 and 0.25);
 \end{tikzpicture}
diff --git a/.local/share/texmf/tex/latex/images/unit-circle.tikz b/.local/share/texmf/tex/latex/images/unit-circle.tikz
@@ -1,7 +1,7 @@
 % This picture represents the unit complex circle
 \begin{tikzpicture}[scale=1.2]
-  \node[above] (O) at (0,1) {\(i\)};
-  \node[right] (O) at (1,0) {\(1\)};
+  \node[above] (O) at (0,1) {$i$};
+  \node[right] (O) at (1,0) {$1$};
   \filldraw[black] (0, 1) circle (1pt);
   \filldraw[black] (1, 0) circle (1pt);
   \draw (0, 0) circle (1);
diff --git a/.local/share/texmf/tex/latex/images/velocity.tikz b/.local/share/texmf/tex/latex/images/velocity.tikz
@@ -1,6 +1,6 @@
 \begin{tikzpicture}[scale=0.7]
   % The manifold
-  \draw (-4, 2) node[below]{\(M\)} to[relative, out=20, in=160] 
+  \draw (-4, 2) node[below]{$M$} to[relative, out=20, in=160] 
         ( 2, 0) to[relative, out=20, in=160] 
         ( 4, 3) to[relative, out=-20, in=-160] cycle;
 
diff --git a/.local/share/texmf/tex/latex/xalgebra.sty b/.local/share/texmf/tex/latex/xalgebra.sty
@@ -22,13 +22,15 @@
 % Linear Algebra stuff
 \newenvironment{system}
   {\left \{ \begin{aligned}}
-  {\end{aligned} \right.}       % Linear system of equations
-\DeclareMathOperator{\Tr}{Tr}   % Operator trace
-\DeclareMathOperator{\Id}{Id}   % Identity operator
-\DeclareMathOperator{\Bil}{Bil} % The space of bilinear maps
-\DeclareMathOperator{\Mat}{Mat} % Matrix algebra
-\DeclareMathOperator{\Sym}{Sym} % Symetric product of vector-spaces
-\newcommand{\base}{\mathscr B}  % Vectorspace base
+  {\end{aligned} \right.}         % Linear system of equations
+\DeclareMathOperator{\Tr}{Tr}     % Operator trace
+\DeclareMathOperator{\Id}{Id}     % Identity operator
+\DeclareMathOperator{\Bil}{Bil}   % The space of bilinear maps
+\DeclareMathOperator{\Mat}{Mat}   % Matrix algebra
+\DeclareMathOperator{\Sym}{Sym}   % Symetric product of vector-spaces
+\newcommand{\base}{\mathscr B}    % Vectorspace base
+\renewcommand{\Re}{\mathrm{Re}\,} % Real component
+\renewcommand{\Im}{\mathrm{Im}\,} % Imaginary component
 \newcommand{\norm}[1]{\left\lVert\nobreak#1\nobreak\right\rVert} % Vector norm
 
 % Group Theory stuff