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LaTeX Coordinate Transformations

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Sep 23rd, 2017
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  1. \documentclass[a4paper]{scrreprt}
  2.  
  3. \usepackage%
  4. [%
  5.     a4paper,                % Paper format
  6.     textwidth = 160.0mm,    % Text width
  7.     textheight = 255.0mm,   % Text height
  8.     bindingoffset = 0mm,    % No offset for the margins
  9.     inner = 32.5mm,         % Inner margin (Middle: 25mm)
  10.     ignoreheadfoot,         % Disregarding the header and footer for determining the vertical position of the text body
  11.     top = 25.0mm,           % Distance between the top of the page and the top of the text body
  12.     headheight = 7.5mm,     % Header height
  13.     headsep = 10.0mm,       % Distance between the bottom of the header and the top of the text body
  14.     footskip = 10.0mm       % Distance between the bottom of the text and the first footer line
  15. ]{geometry}
  16.  
  17. \usepackage{xcolor}
  18. \definecolor{GreenDark}{RGB}{000,179,000}
  19. \newcommand{\VL}[1]{\mathbf{#1}}
  20.  
  21. \usepackage{tikz}
  22. \usepackage{tikz-3dplot}
  23. \usepackage{pgfplots}
  24. \pgfplotsset{compat = 1.3}
  25. \usetikzlibrary{arrows}
  26. \usetikzlibrary{calc}
  27. \newcommand{\TikzInnerSep}{0.33mm}
  28.  
  29. % Define new arc command
  30. % Syntax: [draw options] (center) (initial angle:final angle:radius)
  31. % https://tex.stackexchange.com/a/66220
  32. \def\centerarc[#1](#2)(#3:#4:#5){%
  33.     \draw [#1] ($(#2)+({#5*cos(#3)}, {#5*sin(#3)})$) arc (#3:#4:#5)%
  34. }
  35.  
  36. % Redefine the rotation sequence for the tikz3d-plot to Euler-Angles:
  37. % z-x-z with alpha-beta-gamma (psi-theta-phi)
  38. % https://tex.stackexchange.com/q/118069/98906
  39. \newcommand{\tdseteulerzxz}{%
  40.     \renewcommand{\tdplotcalctransformrotmain}{%
  41.         %
  42.         % Determine the sin and cos of the specified angle in degrees
  43.         % \tdplotsinandcos{sin}{cos}{theta}
  44.         % - #1: Returns sin(#3)
  45.         % - #2: Returns cos(#3)
  46.         % - #3: User-specified angle theta
  47.         \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha}
  48.         \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
  49.         \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
  50.         %
  51.         % Define trigonometric abbreviations
  52.         %
  53.         \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
  54.         \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
  55.         \tdplotmult{\sacbsg}{\sacb}{\singamma}
  56.         \tdplotmult{\sacbcg}{\sacb}{\cosgamma}
  57.         \tdplotmult{\sasg}{\sinalpha}{\singamma}
  58.         \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
  59.         %
  60.         \tdplotmult{\sbsg}{\sinbeta}{\singamma}
  61.         \tdplotmult{\sbcg}{\sinbeta}{\cosgamma}
  62.         %
  63.         \tdplotmult{\casb}{\cosalpha}{\sinbeta}
  64.         \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
  65.         \tdplotmult{\cacbsg}{\cacb}{\singamma}
  66.         \tdplotmult{\cacbcg}{\cacb}{\cosgamma}
  67.         \tdplotmult{\casg}{\cosalpha}{\singamma}
  68.         \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
  69.         %
  70.         % Define the entries for the rotation matrix from the B-System to the I-System
  71.         % This is A_IB = (A_BI)^T
  72.         %
  73.         \pgfmathsetmacro{\raaeul}{+\cacg - \sacbsg}
  74.         \pgfmathsetmacro{\rabeul}{-\casg - \sacbcg}
  75.         \pgfmathsetmacro{\raceul}{+\sasb}
  76.         %
  77.         \pgfmathsetmacro{\rbaeul}{+\sacg + \cacbsg}
  78.         \pgfmathsetmacro{\rbbeul}{-\sasg + \cacbcg}
  79.         \pgfmathsetmacro{\rbceul}{-\casb}
  80.         %
  81.         \pgfmathsetmacro{\rcaeul}{+\sbsg}
  82.         \pgfmathsetmacro{\rcbeul}{+\sbcg}
  83.         \pgfmathsetmacro{\rcceul}{+\cosbeta}
  84.         %
  85.     }
  86. }
  87.  
  88. % Redefine the rotation sequence for the tikz3d-plot to Cardan-Angles:
  89. % x-y-z with alpha-beta-gamma
  90. % https://tex.stackexchange.com/q/118069/98906
  91. \newcommand{\tdsetcardanxyz}{%
  92.     \renewcommand{\tdplotcalctransformrotmain}{%
  93.         %
  94.         % Determine the sin and cos of the specified angle in degrees
  95.         % \tdplotsinandcos{sin}{cos}{theta}
  96.         % - #1: Returns sin(#3)
  97.         % - #2: Returns cos(#3)
  98.         % - #3: User-specified angle theta
  99.         \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha}
  100.         \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
  101.         \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
  102.         %
  103.         % Define trigonometric abbreviations
  104.         %
  105.         \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
  106.         \tdplotmult{\sasbsg}{\sasb}{\singamma}
  107.         \tdplotmult{\sasbcg}{\sasb}{\cosgamma}
  108.         %
  109.         \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
  110.         \tdplotmult{\sasg}{\sinalpha}{\singamma}
  111.         \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
  112.         %
  113.         \tdplotmult{\casb}{\cosalpha}{\sinbeta}
  114.         \tdplotmult{\casbsg}{\casb}{\singamma}
  115.         \tdplotmult{\casbcg}{\casb}{\cosgamma}
  116.         %
  117.         \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
  118.         \tdplotmult{\casg}{\cosalpha}{\singamma}
  119.         \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
  120.         \tdplotmult{\cbsg}{\cosbeta}{\singamma}
  121.         \tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
  122.         %
  123.         % Define the entries for the rotation matrix from the B-System to the I-System
  124.         % This is A_IB = (A_BI)^T
  125.         %
  126.         \pgfmathsetmacro{\raaeul}{+\cbcg}
  127.         \pgfmathsetmacro{\rabeul}{-\cbsg}
  128.         \pgfmathsetmacro{\raceul}{+\sinbeta}
  129.         %
  130.         \pgfmathsetmacro{\rbaeul}{+\casg + \sasbcg}
  131.         \pgfmathsetmacro{\rbbeul}{+\cacg - \sasbsg}
  132.         \pgfmathsetmacro{\rbceul}{-\sacb}
  133.         %
  134.         \pgfmathsetmacro{\rcaeul}{+\sasg - \casbcg}
  135.         \pgfmathsetmacro{\rcbeul}{+\sacg + \casbsg}
  136.         \pgfmathsetmacro{\rcceul}{+\cacb}
  137.         %
  138.     }
  139. }
  140.  
  141. % Plot display orientation
  142. \tdplotsetmaincoords{70}{150}
  143.  
  144. % Rotation angles Euler
  145. \pgfmathsetmacro{\zOneRot}{10}
  146. \pgfmathsetmacro{\xRot}{15}
  147. \pgfmathsetmacro{\zTwoRot}{30}
  148.  
  149. % Rotation angles Cardan
  150. \pgfmathsetmacro{\xRot}{10}
  151. \pgfmathsetmacro{\yRot}{15}
  152. \pgfmathsetmacro{\zRot}{30}
  153.  
  154. \begin{document}
  155.  
  156. \chapter{Euler}
  157.  
  158. % ====================================================================================================================================================
  159. % Euler FoR I
  160. % ====================================================================================================================================================
  161. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  162.  
  163.     % Set origin
  164.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  165.     \coordinate (Origin) at (0, 0, 0);
  166.    
  167.     % Draw grid for Inertial FoR I
  168.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  169.     %
  170.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  171.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  172.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  173.  
  174.     % Inertial FoR I
  175.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  176.     %
  177.     % x_I
  178.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  179.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  180.             {$\VL{e}_{x,I}$};
  181.     %
  182.     % y_I
  183.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  184.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  185.             {$\VL{e}_{y,I}$};
  186.     %
  187.     % z_I
  188.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  189.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  190.             {$\VL{e}_{z,I}$};
  191.  
  192. \end{tikzpicture}
  193. %
  194. \qquad\qquad
  195. %
  196. % ====================================================================================================================================================
  197. % Euler FoR I, FoR 1
  198. % ====================================================================================================================================================
  199. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  200.  
  201.     % Set origin
  202.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  203.     \coordinate (Origin) at (0, 0, 0);
  204.    
  205.     % Draw grid for Inertial FoR I (xy and xz)
  206.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  207.     %
  208.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  209.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  210.  
  211.     % Inertial FoR I
  212.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  213.     %
  214.     % x_I
  215.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  216.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  217.             {$\VL{e}_{x,I}$};
  218.     %
  219.     % y_I
  220.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  221.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  222.             %{$\VL{e}_{y,I}$};
  223.     %
  224.     % z_I
  225.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  226.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  227.             %{$\VL{e}_{z,I}$};
  228.        
  229.     % Intermediate FoR 1
  230.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  231.     %
  232.     % Set the FoR orientation
  233.     \tdseteulerzxz
  234.     \tdplotsetrotatedcoords{\zOneRot}{0}{0}
  235.     %
  236.     % x_1
  237.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  238.         node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  239.             {$\VL{e}_{x,1}$};
  240.     %
  241.     % y_1
  242.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  243.         node[anchor = west, xshift = +1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  244.             {$\VL{e}_{y,1}$};
  245.     %
  246.     % z_1
  247.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  248.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  249.             %{$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  250.    
  251.     % Draw grid for Inertial FoR I (yz) and planes for FoR 1
  252.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  253.     %
  254.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  255.     %
  256.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  257.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  258.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  259.    
  260.     % Draw selected axes over the grid
  261.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  262.     %
  263.     % y_I
  264.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  265.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  266.             {$\VL{e}_{y,I}$};
  267.     %
  268.     % z_1
  269.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  270.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  271.             {$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  272.    
  273.     % Draw rotation arrow
  274.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  275.     %
  276.     \centerarc[tdplot_rotated_coords, canvas is yx plane at z=0.7, arrows = {latex}-{}](0, 0)(230 : 570 : 0.15)%
  277.         node[anchor = north west, xshift = +2mm, yshift = -3mm, inner sep = \TikzInnerSep]%
  278.             {1. $\psi$};
  279.  
  280. \end{tikzpicture}
  281. %
  282. \\[5mm]
  283. %
  284. % ====================================================================================================================================================
  285. % Euler FoR I, FoR 1, FoR 2
  286. % ====================================================================================================================================================
  287. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  288.  
  289.     % Set origin
  290.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  291.     \coordinate (Origin) at (0, 0, 0);
  292.    
  293.     % Draw grid for Inertial FoR I (xy)
  294.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  295.     %
  296.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  297.  
  298.     % Inertial FoR I
  299.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  300.     %
  301.     % x_I
  302.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  303.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  304.             %{$\VL{e}_{x,I}$};
  305.     %
  306.     % y_I
  307.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  308.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  309.             %{$\VL{e}_{y,I}$};
  310.     %
  311.     % z_I
  312.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  313.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  314.             %{$\VL{e}_{z,I}$};
  315.        
  316.     % Intermediate FoR 1
  317.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  318.     %
  319.     % Set the FoR orientation
  320.     \tdseteulerzxz
  321.     \tdplotsetrotatedcoords{\zOneRot}{0}{0}
  322.     %
  323.     % x_1
  324.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  325.         %node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  326.             %{$\VL{e}_{x,1}$};
  327.     %
  328.     % y_1
  329.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  330.         node[anchor = west, xshift = +1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  331.             {$\VL{e}_{y,1}$};
  332.     %
  333.     % z_1
  334.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  335.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  336.             %{$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  337.    
  338.     % Intermediate FoR 2
  339.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  340.     %
  341.     % Set the FoR orientation
  342.     \tdseteulerzxz
  343.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{0}
  344.     %
  345.     % x_2
  346.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  347.         %node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  348.             %{$\textcolor{blue}{\VL{e}_{x,1}} \; \textcolor{black}{=} \; \VL{e}_{x,2}$};
  349.     %
  350.     % y_2
  351.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 1, 0)%
  352.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  353.             {$\VL{e}_{y,2}$};
  354.     %
  355.     % z_2
  356.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 0, 1)%
  357.         node[anchor = east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  358.             {$\VL{e}_{z,2}$};
  359.    
  360.     % Draw grid for Inertial FoR I (xz and yz) and planes for FoR 2
  361.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  362.     %
  363.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  364.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  365.     %
  366.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  367.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  368.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  369.    
  370.     % Draw selected axes over the grid
  371.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  372.     %
  373.     % x_I
  374.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  375.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  376.             {$\VL{e}_{x,I}$};
  377.     %
  378.     % y_I
  379.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  380.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  381.             {$\VL{e}_{y,I}$};
  382.     %
  383.     % Set the FoR orientation
  384.     \tdseteulerzxz
  385.     \tdplotsetrotatedcoords{\zOneRot}{0}{0}
  386.     %
  387.     % z_1
  388.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  389.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  390.             {$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  391.     %
  392.     % Set the FoR orientation
  393.     \tdseteulerzxz
  394.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{0}
  395.     %
  396.     % x_2
  397.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  398.         node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  399.             {$\textcolor{blue}{\VL{e}_{x,1}} \; \textcolor{black}{=} \; \VL{e}_{x,2}$};
  400.    
  401.     % Draw rotation arrow
  402.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  403.     %
  404.     \centerarc[tdplot_rotated_coords, canvas is zy plane at x=0.7, arrows = {latex}-{}](0, 0)(265 : 605 : 0.15)%
  405.         node[anchor = south, xshift = +1mm, yshift = +7mm, inner sep = \TikzInnerSep]%
  406.             {2. $\vartheta$};
  407.  
  408. \end{tikzpicture}
  409. %
  410. \qquad\qquad
  411. %
  412. % ====================================================================================================================================================
  413. % Euler FoR I, FoR 1, FoR 2, FoR B
  414. % ====================================================================================================================================================
  415. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  416.  
  417.     % Set origin
  418.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  419.     \coordinate (Origin) at (0, 0, 0);
  420.    
  421.     % Draw grid for Inertial FoR I (xy)
  422.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  423.     %
  424.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  425.  
  426.     % Inertial FoR I
  427.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  428.     %
  429.     % x_I
  430.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  431.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  432.             %{$\VL{e}_{x,I}$};
  433.     %
  434.     % y_I
  435.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  436.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  437.             %{$\VL{e}_{y,I}$};
  438.     %
  439.     % z_I
  440.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  441.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  442.             %{$\VL{e}_{z,I}$};
  443.        
  444.     % Intermediate FoR 1
  445.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  446.     %
  447.     % Set the FoR orientation
  448.     \tdseteulerzxz
  449.     \tdplotsetrotatedcoords{\zOneRot}{0}{0}
  450.     %
  451.     % x_1
  452.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  453.         %node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  454.             %{$\VL{e}_{x,1}$};
  455.     %
  456.     % y_1
  457.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  458.         node[anchor = west, xshift = +1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  459.             {$\VL{e}_{y,1}$};
  460.     %
  461.     % z_1
  462.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  463.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  464.             %{$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  465.    
  466.     % Intermediate FoR 2
  467.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  468.     %
  469.     % Set the FoR orientation
  470.     \tdseteulerzxz
  471.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{0}
  472.     %
  473.     % x_2
  474.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  475.         %node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  476.             %{$\textcolor{blue}{\VL{e}_{x,1}} \; \textcolor{black}{=} \; \VL{e}_{x,2}$};
  477.     %
  478.     % y_2
  479.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 1, 0)%
  480.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  481.             {$\VL{e}_{y,2}$};
  482.     %
  483.     % z_2
  484.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 0, 1)%
  485.         %node[anchor = east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  486.             %{$\VL{e}_{z,2}$};
  487.    
  488.     % Body-Fixed FoR B
  489.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  490.     %
  491.     % Set the FoR orientation
  492.     \tdseteulerzxz
  493.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{\zTwoRot}
  494.     %
  495.     % x_B
  496.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  497.         %node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  498.             %{$\VL{e}_{x,B}$};
  499.     %
  500.     % y_B
  501.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 1, 0)%
  502.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  503.             {$\VL{e}_{y,B}$};
  504.     %
  505.     % z_B
  506.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 0, 1)%
  507.         node[anchor = east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  508.             {$\textcolor{orange}{\VL{e}_{z,2}} \; \textcolor{black}{=} \; \VL{e}_{z,B}$};
  509.    
  510.     % Draw grid for Inertial FoR I (xz and yz) and planes for FoR B
  511.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  512.     %
  513.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  514.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  515.     %
  516.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  517.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  518.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  519.    
  520.     % Draw selected axes over the grid
  521.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  522.     %
  523.     % x_I
  524.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  525.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  526.             {$\VL{e}_{x,I}$};
  527.     %
  528.     % y_I
  529.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  530.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  531.             {$\VL{e}_{y,I}$};
  532.     %
  533.     % Set the FoR orientation
  534.     \tdseteulerzxz
  535.     \tdplotsetrotatedcoords{\zOneRot}{0}{0}
  536.     %
  537.     % z_1
  538.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  539.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  540.             {$\textcolor{GreenDark}{\VL{e}_{z,I}} \; \textcolor{black}{=} \; \VL{e}_{z,1}$};
  541.     %
  542.     % Set the FoR orientation
  543.     \tdseteulerzxz
  544.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{0}
  545.     %
  546.     % x_2
  547.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  548.         node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  549.             {$\textcolor{blue}{\VL{e}_{x,1}} \; \textcolor{black}{=} \; \VL{e}_{x,2}$};
  550.     %
  551.     % Set the FoR orientation
  552.     \tdseteulerzxz
  553.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{\zTwoRot}
  554.     %
  555.     % x_B
  556.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  557.         node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  558.             {$\VL{e}_{x,B}$};
  559.    
  560.     % Draw rotation arrow
  561.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  562.     %
  563.     \centerarc[tdplot_rotated_coords, canvas is yx plane at z=0.7, arrows = {latex}-{}](0, 0)(255 : 595 : 0.15)%
  564.         node[anchor = north east, xshift = +0mm, yshift = -2mm, inner sep = \TikzInnerSep]%
  565.             {3. $\varphi$};
  566.  
  567. \end{tikzpicture}
  568. %
  569. \\[5mm]
  570. %
  571. % ====================================================================================================================================================
  572. % Euler FoR I, FoR B
  573. % ====================================================================================================================================================
  574. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  575.  
  576.     % Set origin
  577.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  578.     \coordinate (Origin) at (0, 0, 0);
  579.    
  580.     % Draw grid for Inertial FoR I (xy)
  581.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  582.     %
  583.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  584.  
  585.     % Inertial FoR I
  586.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  587.     %
  588.     % x_I
  589.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  590.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  591.             %{$\VL{e}_{x,I}$};
  592.     %
  593.     % y_I
  594.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  595.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  596.             %{$\VL{e}_{y,I}$};
  597.     %
  598.     % z_I
  599.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  600.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  601.             %{$\VL{e}_{z,I}$};
  602.    
  603.     % Body-Fixed FoR B
  604.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  605.     %
  606.     % Set the FoR orientation
  607.     \tdseteulerzxz
  608.     \tdplotsetrotatedcoords{\zOneRot}{\xRot}{\zTwoRot}
  609.     %
  610.     % x_B
  611.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  612.         node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  613.             {$\VL{e}_{x,B}$};
  614.     %
  615.     % y_B
  616.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 1, 0)%
  617.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  618.             {$\VL{e}_{y,B}$};
  619.     %
  620.     % z_B
  621.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 0, 1)%
  622.         node[anchor = east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  623.             {$\VL{e}_{z,B}$};
  624.    
  625.     % Draw grid for Inertial FoR I (xz and yz)
  626.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  627.     %
  628.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  629.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  630.    
  631.     % Draw selected axes over the grid
  632.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  633.     %
  634.     % x_I
  635.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  636.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  637.             {$\VL{e}_{x,I}$};
  638.     %
  639.     % y_I
  640.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  641.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  642.             {$\VL{e}_{y,I}$};
  643.     %
  644.     % z_I
  645.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  646.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  647.             {$\VL{e}_{z,I}$};
  648.     %
  649.     % x_B
  650.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  651.         node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  652.             {$\VL{e}_{x,B}$};
  653.  
  654. \end{tikzpicture}
  655.  
  656. \newpage\noindent
  657. \chapter{Cardan}
  658.  
  659. % ====================================================================================================================================================
  660. % Cardan FoR I
  661. % ====================================================================================================================================================
  662. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  663.  
  664.     % Set origin
  665.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  666.     \coordinate (Origin) at (0, 0, 0);
  667.    
  668.     % Draw grid for Inertial FoR I
  669.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  670.     %
  671.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  672.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  673.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  674.  
  675.     % Inertial FoR I
  676.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  677.     %
  678.     % x_I
  679.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  680.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  681.             {$\VL{e}_{x,I}$};
  682.     %
  683.     % y_I
  684.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  685.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  686.             {$\VL{e}_{y,I}$};
  687.     %
  688.     % z_I
  689.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  690.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  691.             {$\VL{e}_{z,I}$};
  692.  
  693. \end{tikzpicture}
  694. %
  695. \qquad\qquad
  696. %
  697. % ====================================================================================================================================================
  698. % Cardan FoR I, FoR 1
  699. % ====================================================================================================================================================
  700. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  701.  
  702.     % Set origin
  703.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  704.     \coordinate (Origin) at (0, 0, 0);
  705.    
  706.     % Draw grid for Inertial FoR I (xy and yz)
  707.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  708.     %
  709.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  710.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  711.  
  712.     % Inertial FoR I
  713.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  714.     %
  715.     % x_I
  716.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  717.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  718.             %{$\VL{e}_{x,I}$};
  719.     %
  720.     % y_I
  721.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  722.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  723.             {$\VL{e}_{y,I}$};
  724.     %
  725.     % z_I
  726.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  727.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  728.             {$\VL{e}_{z,I}$};
  729.        
  730.     % Intermediate FoR 1
  731.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  732.     %
  733.     % Set the FoR orientation
  734.     \tdsetcardanxyz
  735.     \tdplotsetrotatedcoords{\xRot}{0}{0}
  736.     %
  737.     % x_1
  738.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  739.         %node[anchor = east, inner sep = \TikzInnerSep]%
  740.             %{$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  741.     %
  742.     % y_1
  743.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  744.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  745.             {$\VL{e}_{y,1}$};
  746.     %
  747.     % z_1
  748.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  749.         node[anchor = south, xshift = -1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  750.             {$\VL{e}_{z,1}$};
  751.    
  752.     % Draw grid for Inertial FoR I (xz) and planes for FoR 1
  753.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  754.     %
  755.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  756.     %
  757.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  758.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  759.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = blue, draw = black] (0, 0) rectangle (1, 1);
  760.    
  761.     % Draw selected axes over the grid
  762.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  763.     %
  764.     % z_I
  765.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  766.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  767.             {$\VL{e}_{z,I}$};
  768.     %
  769.     % x_1
  770.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  771.         node[anchor = east, inner sep = \TikzInnerSep]%
  772.             {$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  773.    
  774.     % Draw rotation arrow
  775.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  776.     %
  777.     \centerarc[tdplot_rotated_coords, canvas is zy plane at x=0.7, arrows = {latex}-{}](0, 0)(265 : 605 : 0.15)%
  778.         node[anchor = north west, xshift = +1mm, yshift = -3mm, inner sep = \TikzInnerSep]%
  779.             {1. $\alpha$};
  780.  
  781. \end{tikzpicture}
  782. %
  783. \\[5mm]
  784. %
  785. % ====================================================================================================================================================
  786. % Cardan FoR I, FoR 1, FoR 2
  787. % ====================================================================================================================================================
  788. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  789.  
  790.     % Set origin
  791.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  792.     \coordinate (Origin) at (0, 0, 0);
  793.    
  794.     % Draw grid for Inertial FoR I (yz)
  795.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  796.     %
  797.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  798.  
  799.     % Inertial FoR I
  800.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  801.     %
  802.     % x_I
  803.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  804.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  805.             %{$\VL{e}_{x,I}$};
  806.     %
  807.     % y_I
  808.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  809.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  810.             %{$\VL{e}_{y,I}$};
  811.     %
  812.     % z_I
  813.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  814.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  815.             %{$\VL{e}_{z,I}$};
  816.        
  817.     % Intermediate FoR 1
  818.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  819.     %
  820.     % Set the FoR orientation
  821.     \tdsetcardanxyz
  822.     \tdplotsetrotatedcoords{\xRot}{0}{0}
  823.     %
  824.     % x_1
  825.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  826.         %node[anchor = east, inner sep = \TikzInnerSep]%
  827.             %{$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  828.     %
  829.     % y_1
  830.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  831.         %node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  832.             %{$\VL{e}_{y,1}$};
  833.     %
  834.     % z_1
  835.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  836.         node[anchor = south, xshift = -1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  837.             {$\VL{e}_{z,1}$};
  838.    
  839.     % Intermediate FoR 2
  840.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  841.     %
  842.     % Set the FoR orientation
  843.     \tdsetcardanxyz
  844.     \tdplotsetrotatedcoords{\xRot}{\yRot}{0}
  845.     %
  846.     % x_2
  847.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  848.         node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  849.             {$\VL{e}_{x,2}$};
  850.     %
  851.     % y_2
  852.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 1, 0)%
  853.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  854.             {$\textcolor{blue}{\VL{e}_{y,1}} \; \textcolor{black}{=} \; \VL{e}_{y,2}$};
  855.     %
  856.     % z_2
  857.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 0, 1)%
  858.         node[anchor = east, xshift = -1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  859.             {$\VL{e}_{z,2}$};
  860.    
  861.     % Draw grid for Inertial FoR I (xy and xz) and planes for FoR 2
  862.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  863.     %
  864.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  865.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  866.     %
  867.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  868.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  869.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = orange, draw = black] (0, 0) rectangle (1, 1);
  870.  
  871.     % Draw selected axes over the grid
  872.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  873.     %
  874.     % y_I
  875.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  876.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  877.             {$\VL{e}_{y,I}$};
  878.     %
  879.     % z_I
  880.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  881.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  882.             {$\VL{e}_{z,I}$};
  883.     %
  884.     % Set the FoR orientation
  885.     \tdsetcardanxyz
  886.     \tdplotsetrotatedcoords{\xRot}{0}{0}
  887.     %
  888.     % x_1
  889.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  890.         node[anchor = east, inner sep = \TikzInnerSep]%
  891.             {$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  892.    
  893.     % Draw rotation arrow
  894.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  895.     %
  896.     \centerarc[tdplot_rotated_coords, canvas is zx plane at y=0.7, arrows = {}-{latex}](0, 0)(255 : 595 : 0.15)%
  897.         node[anchor = south, xshift = +1mm, yshift = +5mm, inner sep = \TikzInnerSep]%
  898.             {2. $\beta$};
  899.  
  900. \end{tikzpicture}
  901. %
  902. \qquad\qquad
  903. %
  904. % ====================================================================================================================================================
  905. % FoR I, FoR 1, FoR 2, FoR B
  906. % ====================================================================================================================================================
  907. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  908.  
  909.     % Set origin
  910.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  911.     \coordinate (Origin) at (0, 0, 0);
  912.    
  913.     % Inertial FoR I
  914.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  915.     %
  916.     % x_I
  917.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  918.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  919.             %{$\VL{e}_{x,I}$};
  920.     %
  921.     % y_I
  922.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  923.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  924.             %{$\VL{e}_{y,I}$};
  925.     %
  926.     % z_I
  927.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  928.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  929.             %{$\VL{e}_{z,I}$};
  930.        
  931.     % Intermediate FoR 1
  932.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  933.     %
  934.     % Set the FoR orientation
  935.     \tdsetcardanxyz
  936.     \tdplotsetrotatedcoords{\xRot}{0}{0}
  937.     %
  938.     % x_1
  939.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  940.         %node[anchor = east, inner sep = \TikzInnerSep]%
  941.             %{$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  942.     %
  943.     % y_1
  944.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  945.         %node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  946.             %{$\VL{e}_{y,1}$};
  947.     %
  948.     % z_1
  949.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 0, 1)%
  950.         node[anchor = south, xshift = -1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  951.             {$\VL{e}_{z,1}$};
  952.    
  953.     % Intermediate FoR 2
  954.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  955.     %
  956.     % Set the FoR orientation
  957.     \tdsetcardanxyz
  958.     \tdplotsetrotatedcoords{\xRot}{\yRot}{0}
  959.     %
  960.     % x_2
  961.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (1, 0, 0)%
  962.         node[anchor = north, yshift = -1mm, inner sep = \TikzInnerSep]%
  963.             {$\VL{e}_{x,2}$};
  964.     %
  965.     % y_2
  966.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 1, 0)%
  967.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  968.             {$\textcolor{blue}{\VL{e}_{y,1}} \; \textcolor{black}{=} \; \VL{e}_{y,2}$};
  969.     %
  970.     % z_2
  971.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, orange] (Origin) -- (0, 0, 1)%
  972.         %node[anchor = east, xshift = -1mm, yshift = +1mm, inner sep = \TikzInnerSep]%
  973.             %{$\VL{e}_{z,2}$};
  974.    
  975.     % Body-Fixed FoR B
  976.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  977.     %
  978.     % Set the FoR orientation
  979.     \tdsetcardanxyz
  980.     \tdplotsetrotatedcoords{\xRot}{\yRot}{\zRot}
  981.     %
  982.     % x_B
  983.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  984.         %node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  985.             %{$\VL{e}_{x,B}$};
  986.     %
  987.     % y_B
  988.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 1, 0)%
  989.         node[anchor = south, yshift = +1mm, inner sep = \TikzInnerSep]%
  990.             {$\VL{e}_{y,B}$};
  991.     %
  992.     % z_B
  993.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 0, 1)%
  994.         node[anchor = south east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  995.             {$\textcolor{orange}{\VL{e}_{z,2}} \; \textcolor{black}{=} \; \VL{e}_{z,B}$};
  996.    
  997.     % Draw grid for Inertial FoR I and planes for FoR B
  998.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  999.     %
  1000.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1001.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1002.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1003.     %
  1004.     \fill[tdplot_rotated_coords, canvas is yx plane at z=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  1005.     \fill[tdplot_rotated_coords, canvas is zx plane at y=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  1006.     \fill[tdplot_rotated_coords, canvas is zy plane at x=0, opacity = 0.2, fill = red, draw = black] (0, 0) rectangle (1, 1);
  1007.    
  1008.     % Draw selected axes over the grid
  1009.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1010.     %
  1011.     % y_I
  1012.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  1013.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  1014.             {$\VL{e}_{y,I}$};
  1015.     %
  1016.     % z_I
  1017.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  1018.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  1019.             {$\VL{e}_{z,I}$};
  1020.     %
  1021.     % Set the FoR orientation
  1022.     \tdsetcardanxyz
  1023.     \tdplotsetrotatedcoords{\xRot}{0}{0}
  1024.     %
  1025.     % x_1
  1026.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (1, 0, 0)%
  1027.         node[anchor = east, inner sep = \TikzInnerSep]%
  1028.             {$\textcolor{GreenDark}{\VL{e}_{x,I}} \; \textcolor{black}{=} \; \VL{e}_{x,1}$};
  1029.     %
  1030.     % y_1
  1031.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, blue] (Origin) -- (0, 1, 0)%
  1032.         node[anchor = west, xshift = +1mm, inner sep = \TikzInnerSep]%
  1033.             {$\VL{e}_{y,1}$};
  1034.     %
  1035.     % Set the FoR orientation
  1036.     \tdsetcardanxyz
  1037.     \tdplotsetrotatedcoords{\xRot}{\yRot}{\zRot}
  1038.     %
  1039.     % x_B
  1040.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  1041.         node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  1042.             {$\VL{e}_{x,B}$};
  1043.    
  1044.     % Draw rotation arrow
  1045.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1046.     %
  1047.     \centerarc[tdplot_rotated_coords, canvas is yx plane at z=0.7, arrows = {latex}-{}](0, 0)(245 : 585 : 0.15)%
  1048.         node[anchor = north east, xshift = -1mm, yshift = -3mm, inner sep = \TikzInnerSep]%
  1049.             {3. $\gamma$};
  1050.    
  1051. \end{tikzpicture}
  1052. %
  1053. \\[5mm]
  1054. %
  1055. % ====================================================================================================================================================
  1056. % Cardan FoR I, FoR B
  1057. % ====================================================================================================================================================
  1058. \begin{tikzpicture}[x = 1.0cm, y = 1.0cm, z = 1.0cm, scale = 2.5, tdplot_main_coords]
  1059.  
  1060.     % Set origin
  1061.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1062.     \coordinate (Origin) at (0, 0, 0);
  1063.    
  1064.     % Inertial FoR I
  1065.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1066.     %
  1067.     % x_I
  1068.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  1069.         %node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  1070.             %{$\VL{e}_{x,I}$};
  1071.     %
  1072.     % y_I
  1073.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  1074.         %node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  1075.             %{$\VL{e}_{y,I}$};
  1076.     %
  1077.     % z_I
  1078.     %\draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  1079.         %node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  1080.             %{$\VL{e}_{z,I}$};
  1081.    
  1082.     % Body-Fixed FoR B
  1083.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1084.     %
  1085.     % Set the FoR orientation
  1086.     \tdsetcardanxyz
  1087.     \tdplotsetrotatedcoords{\xRot}{\yRot}{\zRot}
  1088.     %
  1089.     % x_B
  1090.     %\draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  1091.         %node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  1092.             %{$\VL{e}_{x,B}$};
  1093.     %
  1094.     % y_B
  1095.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 1, 0)%
  1096.         node[anchor = south, yshift = +1mm, inner sep = \TikzInnerSep]%
  1097.             {$\VL{e}_{y,B}$};
  1098.     %
  1099.     % z_B
  1100.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (0, 0, 1)%
  1101.         node[anchor = south east, xshift = -1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  1102.             {$\VL{e}_{z,B}$};
  1103.    
  1104.     % Draw grid for Inertial FoR I
  1105.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1106.     %
  1107.     \draw [canvas is yx plane at z=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1108.     \draw [canvas is zx plane at y=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1109.     \draw [canvas is zy plane at x=0, black!10!white, step = 0.2] (0, 0) grid (1, 1);
  1110.    
  1111.     % Draw selected axes over the grid
  1112.     % ------------------------------------------------------------------------------------------------------------------------------------------------
  1113.     %
  1114.     % x_I
  1115.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (1, 0, 0)%
  1116.         node[anchor = east, xshift = -1mm, inner sep = \TikzInnerSep]%
  1117.             {$\VL{e}_{x,I}$};
  1118.     %
  1119.     % y_I
  1120.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 1, 0)%
  1121.         node[anchor = north west, xshift = +1mm, inner sep = \TikzInnerSep]%
  1122.             {$\VL{e}_{y,I}$};
  1123.     %
  1124.     % z_I
  1125.     \draw [arrows = {}-{latex}, line width = 1.0pt, GreenDark] (Origin) -- (0, 0, 1)%
  1126.         node[anchor = south west, xshift = +1mm, yshift = -1mm, inner sep = \TikzInnerSep]%
  1127.             {$\VL{e}_{z,I}$};
  1128.     %
  1129.     % Set the FoR orientation
  1130.     \tdsetcardanxyz
  1131.     \tdplotsetrotatedcoords{\xRot}{\yRot}{\zRot}
  1132.     %
  1133.     % x_B
  1134.     \draw [tdplot_rotated_coords, arrows = {}-{latex}, line width = 1.0pt, red] (Origin) -- (1, 0, 0)%
  1135.         node[anchor = north west, yshift = -1mm, inner sep = \TikzInnerSep]%
  1136.             {$\VL{e}_{x,B}$};
  1137.  
  1138. \end{tikzpicture}
  1139.  
  1140. \end{document}
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