-
Notifications
You must be signed in to change notification settings - Fork 0
/
finiterotation.tex
140 lines (117 loc) · 7.24 KB
/
finiterotation.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
\begin{figure}[!htbp]
\vspace*{-0.5em}\[
\mathcircabove{\bm{e}}_i = \mathcircabove{\bm{e}}_i \dotp \bm{e}_j \bm{e}_j
\]
\vspace{-1.5em}\[
\left[ \begin{array}{c} \mathcircabove{\bm{e}}_1 \\ \mathcircabove{\bm{e}}_2 \\ \mathcircabove{\bm{e}}_3 \end{array} \right] =
\left[ \begin{array}{ccc}
\mathcircabove{\bm{e}}_1 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_1 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_1 \dotp \bm{e}_3 \\
\mathcircabove{\bm{e}}_2 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_2 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_2 \dotp \bm{e}_3 \\
\mathcircabove{\bm{e}}_3 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_3 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_3 \dotp \bm{e}_3
\end{array} \right] \hspace{-0.5ex}
\left[ \hspace{-0.12ex} \begin{array}{c} {\bm{e}}_1 \\ {\bm{e}}_2 \\ {\bm{e}_3} \end{array} \right]
\]
\vspace{-1.25em}
\begin{center}
\tdplotsetmaincoords{60}{120} % set orientation of axes
\pgfmathsetmacro{\angletheta}{42}
% three parameters for vector
\pgfmathsetmacro{\lengthofvector}{0.55}
\pgfmathsetmacro{\anglefromz}{40}
\pgfmathsetmacro{\anglefromx}{240}
\begin{tikzpicture}[scale=4, tdplot_main_coords] % tdplot_main_coords style to use 3dplot
\coordinate (O) at (0,0,0);
% draw initial axes
\draw [line width=1.2pt, black, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (1,0,0)
node[pos=0.9, above, xshift=-0.8em] {$\mathcircabove{\bm{e}}_1$};
\draw [line width=1.2pt, black, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (0,1,0)
node[pos=0.9, above, xshift=1em, yshift=-0.2em] {$\mathcircabove{\bm{e}}_2$};
\draw [line width=1.2pt, red, -{Stealth[round,length=4mm,width=2.4mm]}]
(O) -- (0,0,0.9)
node[anchor=south] {$\mathcircabove{\bm{e}}_3 = \bm{e}_3 = \bm{k}$};
% draw initial vector
\tdplotsetcoord{point}{\lengthofvector}{\anglefromz}{\anglefromx} % {length}{angle from z}{angle from x}
% it also defines (pointxy), (pointxz), and (pointyz) projections of point
\draw [line width=1.2pt, black, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (point)
node[anchor=south] {$\initiallocationvector$};
% draw its projection on xy plane
\draw [line width=0.4pt, dotted, color=black] (O) -- (pointxy);
\draw [line width=0.4pt, dotted, color=black] (pointxy) -- (point);
% draw the angle, and label it
% syntax: \tdplotdrawarc[coordinate frame, draw options]{center point}{r}{angle}{end angle}{label options}{label}
\tdplotdrawarc [line width=0.5pt, red, ->]
{(O)}{0.4}{0}{\angletheta}{anchor=north}{$\vartheta$}
\tdplotdrawarc [line width=0.5pt, red, ->]
{(O)}{0.4}{90}{90+\angletheta}{anchor=west}{$\vartheta$}
% rotate coordinates using Euler angles "z(\alpha)y(\beta)z(\gamma)"
\tdplotsetrotatedcoords{\angletheta}{0}{0}
% draw rotated axes
\draw [line width=1.2pt, blue, tdplot_rotated_coords, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (1,0,0)
node[pos=0.9, left, xshift=-0.1em] {$\bm{e}_1$};
\draw [line width=1.2pt, blue, tdplot_rotated_coords, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (0,1,0)
node[pos=0.9, above, xshift=0.2em, yshift=0.2em] {$\bm{e}_2$};
%%\draw [line width=1.2pt, blue, tdplot_rotated_coords, -{Stealth[round, length=4mm, width=2.4mm]}]
%%(O) -- (0,0,0.8) ;
% draw rotated vector
\tdplotsetcoord{rotatedpoint}%
{\lengthofvector}{\anglefromz}{\anglefromx+\angletheta}
\draw [line width=1.2pt, blue, tdplot_rotated_coords, -{Stealth[round, length=4mm, width=2.4mm]}]
(O) -- (rotatedpoint)
node[anchor=south] {$\currentlocationvector$};
% draw its projection on xy plane
\draw [line width=0.4pt, dotted, color=blue, tdplot_rotated_coords] (O) -- (rotatedpointxy);
\draw [line width=0.4pt, dotted, color=blue, tdplot_rotated_coords] (rotatedpointxy) -- (rotatedpoint);
\tdplotdrawarc [line width=0.5pt, red, ->]
{(O)}{0.28}{\anglefromx}{\anglefromx+\angletheta}{anchor=south east, xshift=0.3em, yshift=-0.1em}{$\vartheta$}
\end{tikzpicture}
\end{center}
\vspace{-1em}\[
\scalebox{0.8}[0.85]{$\left[ \begin{array}{ccc}
\mathcircabove{\bm{e}}_1 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_1 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_1 \dotp \bm{e}_3 \\
\mathcircabove{\bm{e}}_2 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_2 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_2 \dotp \bm{e}_3 \\
\mathcircabove{\bm{e}}_3 \dotp \bm{e}_1 & \mathcircabove{\bm{e}}_3 \dotp \bm{e}_2 & \mathcircabove{\bm{e}}_3 \dotp \bm{e}_3
\end{array} \right]$} \hspace{-0.32ex} = \hspace{-0.2ex}
%
\scalebox{0.8}[0.85]{$\left[ \hspace{-0.2ex} \begin{array}{ccc}
\cosine \vartheta & \hspace{-1ex} \cosine \left( 90\degree \!+ \vartheta \right) & \cosine 90\degree \\
\cosine \left( 90\degree \!- \vartheta \right) & \cosine \vartheta & \cosine 90\degree \\
\cosine 90\degree & \cosine 90\degree & \cosine 0\degree
\end{array} \right]$} \hspace{-0.32ex} = \hspace{-0.2ex}
%
\scalebox{0.8}[0.85]{$\left[ \hspace{-0.1ex} \begin{array}{ccc}
\cosine \vartheta & - \sine \vartheta & 0 \\
\sine \vartheta & \cosine \vartheta & 0 \\
0 & 0 & 1
\end{array} \right]$}
\]
\vspace{-0.8em}
\[\begin{array}{c}
\mathcircabove{\bm{e}}_1 \hspace{-0.16ex} = \bm{e}_1 \cosine \vartheta \hspace{.1ex} - \hspace{.1ex} \bm{e}_2 \sine \vartheta \\[0.1em]
\mathcircabove{\bm{e}}_2 \hspace{-0.16ex} = \bm{e}_1 \sine \vartheta \hspace{.1ex} + \hspace{.1ex} \bm{e}_2 \cosine \vartheta \\[0.1em]
\mathcircabove{\bm{e}}_3 \hspace{-0.16ex} = \bm{e}_3 = \bm{k}
\end{array}\]
\vspace{-1em}
\begin{multline*}
\shoveleft{ \rotationtensor = \bm{e}_1 \hspace{-0.1ex} \mathcircabove{\bm{e}}_1 + \bm{e}_2 \mathcircabove{\bm{e}}_2 + \bm{e}_3 \mathcircabove{\bm{e}}_3 = \hfill }\\[1.5em]
%
= \hspace{.2ex} \tikzmark{StartBraceE1E1} {\bm{e}_1 \bm{e}_1 \cosine \vartheta - \bm{e}_1 \bm{e}_2 \sine \vartheta \hspace{.2em}} \tikzmark{EndBraceE1E1} \hspace{-0.1ex} + \hspace{.1ex} \tikzmark{StartBraceE2E2} {\bm{e}_2 \bm{e}_1 \sine \vartheta + \bm{e}_2 \bm{e}_2 \cosine \vartheta \hspace{.2em}} \tikzmark{EndBraceE2E2} \hspace{-0.1ex} + \tikzmark{StartBraceE3E3} {\hspace{.25ex} \bm{k} \bm{k} \hspace{.1ex}} \tikzmark{EndBraceE3E3} \hspace{.1ex}
=\\[0.32em]
%
= \hspace{.1ex} \UnitDyad \cosine \vartheta - \hspace{-0.1ex} \tikzmark{StartBraceKk} {\hspace{.1ex}\bm{e}_3 \bm{e}_3\hspace{.1ex}} \tikzmark{EndBraceKk} \hspace{-0.25ex} \cosine \vartheta \hspace{.1ex} + \tikzmark{StartBraceSkewsymmetry} {\left( \bm{e}_2 \bm{e}_1 - \bm{e}_1 \bm{e}_2 \right)} \tikzmark{EndBraceSkewsymmetry} \sine \vartheta + \bm{k} \bm{k} \hspace{.1ex}
=\\[1.5em]
%
\shoveright{ \hfill = \UnitDyad \cosine \vartheta + \bm{k} \times \hspace{-0.2ex} \UnitDyad \sine \vartheta + \bm{k} \bm{k} \left({1 - \cosine \vartheta}\right) }
\end{multline*}
\AddOverBrace[line width=0.75pt]{StartBraceE1E1}{EndBraceE1E1}{${\scriptstyle \bm{e}_1 \mathcircabove{\bm{e}}_1}$}
\AddOverBrace[line width=0.75pt]{StartBraceE2E2}{EndBraceE2E2}{${\scriptstyle \bm{e}_2 \mathcircabove{\bm{e}}_2}$}
\AddOverBrace[line width=0.75pt]{StartBraceE3E3}{EndBraceE3E3}{${\scriptstyle \bm{e}_3 \mathcircabove{\bm{e}}_3}$}
\AddUnderBrace[line width=0.75pt][-0.1ex,-0.2ex]{StartBraceKk}{EndBraceKk}{${\scriptstyle \bm{k}\bm{k}}$}
\AddUnderBrace[line width=0.75pt][-0.1ex,-0.2ex][xshift=0.4ex]{StartBraceSkewsymmetry}{EndBraceSkewsymmetry}{${\scriptstyle \bm{e}_3 \times \bm{e}_i \bm{e}_i \:=\: \permutationsparitytensor_{3i\hspace{-0.1ex}j} \bm{e}_j \bm{e}_i}$}
\vspace{-0.5em}
\caption{\inquotes{\en{Finite rotation}\ru{Конечный поворот}}}\label{fig:eulerfiniterotation}
\end{figure}