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@ -123,6 +123,12 @@ Lie Groups
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\AAdd}[1]{\mathbf{\mathop{Ad}}{}_{#1}}
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{\mathbf{\mathop{Ad}}{}_{#1}}
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -583,7 +589,7 @@ T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\co
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\end_inset
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This is a bit clumsy, so we resort to a trick: embed the 2D poses in the
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space of
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\begin_inset Formula $3\times3$
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\end_inset
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@ -697,7 +703,7 @@ e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty
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\end_inset
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The series can be similarly defined for square matrices,and the final result
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The series can be similarly defined for square matrices, and the final result
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is that we can write the motion of a robot along a circular trajectory,
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resulting from the 2D twist
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\begin_inset Formula $\xi=(v,\omega)$
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@ -742,8 +748,9 @@ special Euclidean group
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\end_inset
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.
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It is called a Lie group because it is both a manifold and a group, and
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its group operation is smooth when operating on this manifold.
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It is called a Lie group because it is simultaneously a topological group
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and a manifold, which implies that the multiplication and the inversion
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operations are smooth.
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The space of 2D twists, together with a special binary operation to be
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defined below, is called the Lie algebra
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\begin_inset Formula $\setwo$
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@ -778,7 +785,7 @@ A Lie group
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\begin_inset Formula $G$
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\end_inset
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is a manifold that possesses a smooth group operation.
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is a a group and a manifold that possesses a smooth group operation.
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Associated with it is a Lie Algebra
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\begin_inset Formula $\gg$
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\end_inset
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@ -896,8 +903,15 @@ Instead,
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\begin_inset Formula $Z$
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\end_inset
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can be calculated using the Baker-Campbell-Hausdorff (BCH) formula:
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\begin_inset Foot
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can be calculated using the Baker-Campbell-Hausdorff (BCH) formula
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\begin_inset CommandInset citation
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LatexCommand cite
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key "Hall00book"
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\end_inset
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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@ -906,7 +920,7 @@ http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
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\end_inset
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:
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\begin_inset Formula \[
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Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots\]
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@ -929,7 +943,7 @@ For
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\begin_inset Formula $n$
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\end_inset
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-dimensional matrix Lie groups, the Lie algebra
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-dimensional matrix Lie groups, as a vector space the Lie algebra
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\begin_inset Formula $\gg$
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\end_inset
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@ -937,7 +951,7 @@ For
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\begin_inset Formula $\mathbb{R}^{n}$
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\end_inset
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, and we can define the wedge operator
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, and we can define the hat operator
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\begin_inset CommandInset citation
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LatexCommand cite
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after "page 41"
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@ -990,7 +1004,7 @@ where the
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\begin_inset Formula $n\times n$
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\end_inset
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matrices known as the Lie group generators.
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matrices known as Lie group generators.
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The meaning of the map
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\begin_inset Formula $x\rightarrow\xhat$
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\end_inset
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@ -1003,12 +1017,92 @@ where the
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\end_layout
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\begin_layout Subsection
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The Adjoint Map
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Tangent Spaces and the Tangent Bundle
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\end_layout
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\begin_layout Standard
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Below we frequently make use of the equality
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\begin_inset Foot
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The tangent space
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\begin_inset Formula $T_{g}G$
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\end_inset
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at a group element
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\begin_inset Formula $g\in G$
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\end_inset
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is the vector space of tangent vectors at
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\begin_inset Formula $g$
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\end_inset
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.
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The tangent bundle
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\begin_inset Formula $TG$
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\end_inset
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is the set of all tangent vectors
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\begin_inset Formula \[
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TG\define\bigcup_{g\in G}T_{g}G\]
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\end_inset
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A vector field
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\begin_inset Formula $X:G\rightarrow TG$
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\end_inset
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assigns to each element
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\begin_inset Formula $g$
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\end_inset
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as single tangent vector
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\begin_inset Formula $x\in T_{g}G$
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\end_inset
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.
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\end_layout
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\begin_layout Subsection
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The Adjoint Map and Adjoint Representation
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\end_layout
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\begin_layout Standard
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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We do not fully understand this yet!!!
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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The
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\emph on
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adjoint map
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\emph default
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\begin_inset Formula $\AAdd g$
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\end_inset
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maps a group element
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\begin_inset Formula $a\in G$
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\end_inset
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to its conjugate
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\begin_inset Formula $gag^{-1}$
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\end_inset
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by
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\begin_inset Formula $g$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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Below we frequently make use of the equality
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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@ -1019,7 +1113,7 @@ http://en.wikipedia.org/wiki/Exponential_map
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\begin_inset Formula \[
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ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]
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\AAdd ge^{\xhat}=ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]
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\end_inset
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@ -1031,8 +1125,17 @@ where
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\begin_inset Formula $g$
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\end_inset
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, and is called the
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\emph on
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adjoint representation
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\emph default
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.
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The intuitive explanation is that a change
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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The intuitive explanation is that a change
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\begin_inset Formula $\exp\left(\xhat\right)$
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\end_inset
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@ -1049,8 +1152,16 @@ where
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\end_inset
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.
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In the case of a matrix group the ajoint can be written as
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\begin_inset Foot
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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In the case of a matrix group the adjoint can be written as
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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@ -1307,13 +1418,9 @@ all
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\end_inset
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which we can write in terms of
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\begin_inset Formula $\omega$
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\end_inset
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as
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i.e.,
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\begin_inset Formula \[
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\Ad R\omega=\omega\]
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\Ad R\what=\what\]
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\end_inset
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@ -2292,6 +2399,8 @@ The action of
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\begin_inset Formula \[
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\hat{q}=\left[\begin{array}{c}
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q\\
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1\end{array}\right]=\left[\begin{array}{c}
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Rp+t\\
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1\end{array}\right]=\left[\begin{array}{cc}
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R & t\\
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0 & 1\end{array}\right]\left[\begin{array}{c}
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@ -2372,6 +2481,24 @@ v\end{array}\right]\]
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\end_inset
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The inverse action
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\begin_inset Formula $T^{-1}p$
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\end_inset
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is
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\begin_inset Formula \[
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\hat{q}=\left[\begin{array}{c}
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q\\
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1\end{array}\right]=\left[\begin{array}{c}
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R^{T}(p-t)\\
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1\end{array}\right]=\left[\begin{array}{cc}
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R^{T} & -R^{T}t\\
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0 & 1\end{array}\right]\left[\begin{array}{c}
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p\\
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1\end{array}\right]=T^{-1}\hat{p}\]
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\end_inset
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\end_layout
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