gtsam/doc/LieGroups.lyx

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\begin_body

\begin_layout Title
Lie Groups for Beginners
\end_layout

\begin_layout Author
Frank Dellaert
\end_layout

\begin_layout Standard
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\begin_layout Plain Layout
Derivatives
\end_layout

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\begin_layout Standard
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Lie Groups
\end_layout

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SO(2), 1
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\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
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\begin_inset FormulaMacro
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SE(2), 3
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SO(3), 3
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SL(3),8
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\end_layout

\begin_layout Section
Motivation: Rigid Motions in the Plane
\end_layout

\begin_layout Standard
We will start with a small example of a robot moving in a plane, parameterized
 by a 
\emph on
2D pose
\emph default
 
\begin_inset Formula $(x,\, y,\,\theta)$
\end_inset

.
 When we give it a small forward velocity 
\begin_inset Formula $v_{x}$
\end_inset

, we know that the location changes as 
\begin_inset Formula \[
\dot{x}=v_{x}\]

\end_inset

The solution to this trivial differential equation is, with 
\begin_inset Formula $x_{0}$
\end_inset

 the initial 
\begin_inset Formula $x$
\end_inset

-position of the robot,
\begin_inset Formula \[
x_{t}=x_{0}+v_{x}t\]

\end_inset

A similar story holds for translation in the 
\begin_inset Formula $y$
\end_inset

 direction, and in fact for translations in general:
\begin_inset Formula \[
(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0})\]

\end_inset

Similarly for rotation we have 
\begin_inset Formula \[
(x_{t},\, y_{t},\,\theta_{t})=(x_{0},\, y_{0},\,\theta_{0}+\omega t)\]

\end_inset

where 
\begin_inset Formula $\omega$
\end_inset

 is angular velocity, measured in 
\begin_inset Formula $rad/s$
\end_inset

 in counterclockwise direction.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename images/circular.pdf

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
Robot moving along a circular trajectory.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
However, if we combine translation and rotation, the story breaks down!
 We cannot write
\begin_inset Formula \[
(x_{t},\, y_{t},\,\theta_{t})=(x_{0}+v_{x}t,\, y_{0}+v_{y}t,\,\theta_{0}+\omega t)\]

\end_inset

The reason is that, if we move the robot a tiny bit according to the velocity
 vector 
\begin_inset Formula $(v_{x},\, v_{y},\,\omega)$
\end_inset

, we have (to first order)
\begin_inset Formula \[
(x_{\delta},\, y_{\delta},\,\theta_{\delta})=(x_{0}+v_{x}\delta,\, y_{0}+v_{y}\delta,\,\theta_{0}+\omega\delta)\]

\end_inset

but now the robot has rotated, and for the next incremental change, the
 velocity vector would have to be rotated before it can be applied.
 In fact, the robot will move on a 
\emph on
circular
\emph default
 trajectory.
 
\end_layout

\begin_layout Standard
The reason is that 
\emph on
translation and rotation do not commute
\emph default
: if we rotate and then move we will end up in a different place than if
 we moved first, then rotated.
 In fact, someone once said (I forget who, kudos for who can track down
 the exact quote):
\end_layout

\begin_layout Quote
If rotation and translation commuted, we could do all rotations before leaving
 home.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename images/n-steps.pdf

\end_inset


\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:n-step-program"

\end_inset

Approximating a circular trajectory with 
\begin_inset Formula $n$
\end_inset

 steps.
\end_layout

\end_inset


\end_layout

\end_inset

To make progress, we have to be more precise about how the robot behaves.
 Specifically, let us define composition of two poses 
\begin_inset Formula $T_{1}$
\end_inset

 and 
\begin_inset Formula $T_{2}$
\end_inset

 as
\begin_inset Formula \[
T_{1}T_{2}=(x_{1},\, y_{1},\,\theta_{1})(x_{2},\, y_{2},\,\theta_{2})=(x_{1}+\cos\theta_{1}x_{2}-\sin\theta y_{2},\, y_{1}+\sin\theta_{1}x_{2}+\cos\theta_{1}y_{2},\,\theta_{1}+\theta_{2})\]

\end_inset

This is a bit clumsy, so we resort to a trick: embed the 2D poses in the
 space of 
\begin_inset Formula $3\times3$
\end_inset

 matrices, so we can define composition as matrix multiplication:
\begin_inset Formula \[
T_{1}T_{2}=\left[\begin{array}{cc}
R_{1} & t_{1}\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
R_{2} & t_{2}\\
0 & 1\end{array}\right]=\left[\begin{array}{cc}
R_{1}R_{2} & R_{1}t_{2}+t_{1}\\
0 & 1\end{array}\right]\]

\end_inset

where the matrices 
\begin_inset Formula $R$
\end_inset

 are 2D rotation matrices defined as 
\begin_inset Formula \[
R=\left[\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\end{array}\right]\]

\end_inset

Now a 
\begin_inset Quotes eld
\end_inset

tiny
\begin_inset Quotes erd
\end_inset

 motion of the robot can be written as
\begin_inset Formula \[
T(\delta)=\left[\begin{array}{ccc}
\cos\omega\delta & -\sin\omega\delta & v_{x}\delta\\
\sin\omega\delta & \cos\omega\delta & v_{y}\delta\\
0 & 0 & 1\end{array}\right]\approx\left[\begin{array}{ccc}
1 & -\omega\delta & v_{x}\delta\\
\omega\delta & 1 & v_{y}\delta\\
0 & 0 & 1\end{array}\right]=I+\delta\left[\begin{array}{ccc}
0 & -\omega & v_{x}\\
\omega & 0 & v_{y}\\
0 & 0 & 0\end{array}\right]\]

\end_inset

Let us define the 
\emph on
2D twist
\emph default
 vector 
\begin_inset Formula $\xi=(v,\omega)$
\end_inset

, and the matrix above as
\begin_inset Formula \[
\xihat\define\left[\begin{array}{ccc}
0 & -\omega & v_{x}\\
\omega & 0 & v_{y}\\
0 & 0 & 0\end{array}\right]\]

\end_inset

If we wanted 
\begin_inset Formula $t$
\end_inset

 to be large, we could split up 
\begin_inset Formula $t$
\end_inset

 into smaller timesteps, say 
\begin_inset Formula $n$
\end_inset

 of them, and compose them as follows:
\begin_inset Formula \[
T(t)\approx\left(I+\frac{t}{n}\xihat\right)\ldots\mbox{n times}\ldots\left(I+\frac{t}{n}\xihat\right)=\left(I+\frac{t}{n}\xihat\right)^{n}\]

\end_inset

The result is shown in Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:n-step-program"

\end_inset

.
\end_layout

\begin_layout Standard
Of course, the perfect solution would be obtained if we take 
\begin_inset Formula $n$
\end_inset

 to infinity:
\begin_inset Formula \[
T(t)=\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}\]

\end_inset

For real numbers, this series is familiar and is actually a way to compute
 the exponential function:
\begin_inset Formula \[
e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}\]

\end_inset

The series can be similarly defined for square matrices, and the final result
 is that we can write the motion of a robot along a circular trajectory,
 resulting from the 2D twist 
\begin_inset Formula $\xi=(v,\omega)$
\end_inset


\begin_inset Formula $ $
\end_inset

 as the 
\emph on
matrix exponential
\emph default
 of 
\begin_inset Formula $\xihat$
\end_inset

:
\begin_inset Formula \[
T(t)=e^{t\xihat}\define\lim_{n\rightarrow\infty}\left(I+\frac{t}{n}\xihat\right)^{n}=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\xihat^{k}\]

\end_inset

We call this mapping from 2D twists matrices 
\begin_inset Formula $\xihat$
\end_inset

 to 2D rigid transformations the 
\emph on
exponential map.
\end_layout

\begin_layout Standard
The above has all elements of Lie group theory.
 We call the space of 2D rigid transformations, along with the composition
 operation, the 
\emph on
special Euclidean group
\emph default
 
\begin_inset Formula $\SEtwo$
\end_inset

.
 It is called a Lie group because it is simultaneously a topological group
 and a manifold, which implies that the multiplication and the inversion
 operations are smooth.
 The space of 2D twists, together with a special binary operation to be
 defined below, is called the Lie algebra 
\begin_inset Formula $\setwo$
\end_inset

 associated with 
\begin_inset Formula $\SEtwo$
\end_inset

.
 Below we generalize these concepts and then introduce the most commonly
 used Lie groups and their Lie algebras.
\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
Basic Lie Group Concepts
\end_layout

\begin_layout Subsection
A Manifold and a Group
\end_layout

\begin_layout Standard
A Lie group 
\begin_inset Formula $G$
\end_inset

 is a a group and a manifold that possesses a smooth group operation.
 Associated with it is a Lie Algebra 
\begin_inset Formula $\gg$
\end_inset

 which, loosely speaking, can be identified with the tangent space at the
 identity and completely defines how the groups behaves around the identity.
 There is a mapping from 
\begin_inset Formula $\gg$
\end_inset

 back to 
\begin_inset Formula $G$
\end_inset

, called the exponential map
\begin_inset Formula \[
\exp:\gg\rightarrow G\]

\end_inset

which is typically a many-to-one mapping.
 The corresponding inverse can be define locally around the origin and hence
 is a 
\begin_inset Quotes eld
\end_inset

logarithm
\begin_inset Quotes erd
\end_inset

 
\begin_inset Formula \[
\log:G\rightarrow\gg\]

\end_inset

that maps elements in a neighborhood of 
\begin_inset Formula $id$
\end_inset

 in G to an element in 
\begin_inset Formula $\gg$
\end_inset

.
\end_layout

\begin_layout Subsection
Lie Algebra
\end_layout

\begin_layout Standard
The Lie Algebra 
\begin_inset Formula $\gg$
\end_inset

 is called an algebra because it is endowed with a binary operation, the
 Lie bracket 
\begin_inset Formula $[X,Y]$
\end_inset

, the properties of which are closely related to the group operation of
 
\begin_inset Formula $G$
\end_inset

.
 For example, in matrix Lie groups, the Lie bracket is given by 
\begin_inset Formula $[A,B]\define AB-BA$
\end_inset

.
\end_layout

\begin_layout Standard
The relationship of the Lie bracket to the group operation is as follows:
 for commutative Lie groups vector addition 
\begin_inset Formula $X+Y$
\end_inset

 in 
\begin_inset Formula $\gg$
\end_inset

 mimicks the group operation.
 For example, if we have 
\begin_inset Formula $Z=X+Y$
\end_inset

 in 
\begin_inset Formula $\gg$
\end_inset

, when mapped backed to 
\begin_inset Formula $G$
\end_inset

 via the exponential map we obtain 
\begin_inset Formula \[
e^{Z}=e^{X+Y}=e^{X}e^{Y}\]

\end_inset

However, this does 
\emph on
not
\emph default
 hold for non-commutative Lie groups:
\begin_inset Formula \[
Z=\log(e^{X}e^{Y})\neq X+Y\]

\end_inset

Instead, 
\begin_inset Formula $Z$
\end_inset

 can be calculated using the Baker-Campbell-Hausdorff (BCH) formula
\begin_inset CommandInset citation
LatexCommand cite
key "Hall00book"

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
\end_layout

\end_inset

:
\begin_inset Formula \[
Z=X+Y+[X,Y]/2+[X-Y,[X,Y]]/12-[Y,[X,[X,Y]]]/24+\ldots\]

\end_inset

For commutative groups the bracket is zero and we recover 
\begin_inset Formula $Z=X+Y$
\end_inset

.
 For non-commutative groups we can use the BCH formula to approximate it.
\end_layout

\begin_layout Subsection
Exponential Coordinates
\end_layout

\begin_layout Standard
For 
\begin_inset Formula $n$
\end_inset

-dimensional matrix Lie groups, as a vector space the Lie algebra 
\begin_inset Formula $\gg$
\end_inset

 is isomorphic to 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

, and we can define the hat operator 
\begin_inset CommandInset citation
LatexCommand cite
after "page 41"
key "Murray94book"

\end_inset

,
\begin_inset Formula \[
\hat{}:x\in\mathbb{R}^{n}\rightarrow\xhat\in\gg\]

\end_inset

which maps 
\begin_inset Formula $n$
\end_inset

-vectors 
\begin_inset Formula $x\in\mathbb{R}^{n}$
\end_inset

 to elements of 
\begin_inset Formula $\gg$
\end_inset

.
 In the case of matrix Lie groups, the elements 
\begin_inset Formula $\xhat$
\end_inset

 of 
\begin_inset Formula $\gg$
\end_inset

 are 
\begin_inset Formula $n\times n$
\end_inset

 matrices, and the map is given by
\begin_inset Formula \begin{equation}
\xhat=\sum_{i=1}^{n}x_{i}G^{i}\label{eq:generators}\end{equation}

\end_inset

where the 
\begin_inset Formula $G^{i}$
\end_inset

 are 
\begin_inset Formula $n\times n$
\end_inset

 matrices known as Lie group generators.
 The meaning of the map 
\begin_inset Formula $x\rightarrow\xhat$
\end_inset

 will depend on the group 
\begin_inset Formula $G$
\end_inset

 and will generally have an intuitive interpretation.
\end_layout

\begin_layout Subsection
Tangent Spaces and the Tangent Bundle
\end_layout

\begin_layout Standard
The tangent space 
\begin_inset Formula $T_{g}G$
\end_inset

 at a group element 
\begin_inset Formula $g\in G$
\end_inset

 is the vector space of tangent vectors at 
\begin_inset Formula $g$
\end_inset

.
 The tangent bundle 
\begin_inset Formula $TG$
\end_inset

 is the set of all tangent vectors
\begin_inset Formula \[
TG\define\bigcup_{g\in G}T_{g}G\]

\end_inset

A vector field 
\begin_inset Formula $X:G\rightarrow TG$
\end_inset

 assigns to each element 
\begin_inset Formula $g$
\end_inset

 as single tangent vector 
\begin_inset Formula $x\in T_{g}G$
\end_inset

.
\end_layout

\begin_layout Subsection
The Adjoint Map and Adjoint Representation
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
We do not fully understand this yet!!!
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 
\emph on
adjoint map
\emph default
 
\begin_inset Formula $\AAdd g$
\end_inset

 maps a group element 
\begin_inset Formula $a\in G$
\end_inset

 to its conjugate 
\begin_inset Formula $gag^{-1}$
\end_inset

 by 
\begin_inset Formula $g$
\end_inset

.
\end_layout

\begin_layout Standard
Below we frequently make use of the equality 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
http://en.wikipedia.org/wiki/Exponential_map
\end_layout

\end_inset


\begin_inset Formula \[
\AAdd ge^{\xhat}=ge^{\xhat}g^{-1}=e^{\Ad g{\xhat}}\]

\end_inset

where 
\begin_inset Formula $\Ad g:\gg\rightarrow\mathfrak{\gg}$
\end_inset

 is a map parameterized by a group element 
\begin_inset Formula $g$
\end_inset

, and is called the 
\emph on
adjoint representation
\emph default
.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
The intuitive explanation is that a change 
\begin_inset Formula $\exp\left(\xhat\right)$
\end_inset

 defined around the orgin, but applied at the group element 
\begin_inset Formula $g$
\end_inset

, can be written in one step by taking the adjoint 
\begin_inset Formula $\Ad g{\xhat}$
\end_inset

 of 
\begin_inset Formula $\xhat$
\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the case of a matrix group the adjoint can be written as 
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_group
\end_layout

\end_inset


\begin_inset Formula \[
\Ad T{\xhat}\define T\xhat T^{-1}\]

\end_inset

and hence we have
\end_layout

\begin_layout Standard
\begin_inset Formula \[
Te^{\xhat}T^{-1}=e^{T\xhat T^{-1}}\]

\end_inset

where both 
\begin_inset Formula $T\in G$
\end_inset

 and 
\begin_inset Formula $\xhat\in\gg$
\end_inset

 are 
\begin_inset Formula $n\times n$
\end_inset

 matrices for an 
\begin_inset Formula $n$
\end_inset

-dimensional Lie group.
\end_layout

\begin_layout Subsection
Actions
\end_layout

\begin_layout Standard
The (usual) action of an 
\begin_inset Formula $n$
\end_inset

-dimensional matrix group 
\begin_inset Formula $G$
\end_inset

 is matrix-vector multiplication on 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

, 
\begin_inset Formula \[
q=Tp\]

\end_inset

with 
\begin_inset Formula $p,q\in\mathbb{R}^{n}$
\end_inset

 and 
\begin_inset Formula $T\in GL(n)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
2D Rotations
\end_layout

\begin_layout Standard
We first look at a very simple group, the 2D rotations.
\end_layout

\begin_layout Subsection
Basics
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $\SOtwo$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(2)$
\end_inset

 of 
\begin_inset Formula $2\times2$
\end_inset

 invertible matrices.
 Its Lie algebra 
\begin_inset Formula $\sotwo$
\end_inset

 is the vector space of 
\begin_inset Formula $2\times2$
\end_inset

 skew-symmetric matrices.
 Since 
\begin_inset Formula $\SOtwo$
\end_inset

 is a one-dimensional manifold, 
\begin_inset Formula $\sotwo$
\end_inset

 is isomorphic to 
\begin_inset Formula $\mathbb{R}$
\end_inset

 and we define
\begin_inset Formula \[
\hat{}:\mathbb{R}\rightarrow\sotwo\]

\end_inset


\begin_inset Formula \[
\hat{}:\omega\rightarrow\what=\skew{\omega}\]

\end_inset

which maps the angle 
\begin_inset Formula $\theta$
\end_inset

 to the 
\begin_inset Formula $2\times2$
\end_inset

 skew-symmetric matrix 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none

\begin_inset Formula $\skew{\theta}$
\end_inset

:
\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset Formula \[
\skew{\theta}=\left[\begin{array}{cc}
0 & -\theta\\
\theta & 0\end{array}\right]\]

\end_inset

The exponential map can be computed in closed form as
\begin_inset Formula \[
R=e^{\skew{\omega t}}=e^{\skew{\theta}}=\left[\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\end{array}\right]\]

\end_inset

This can be proven 
\begin_inset CommandInset citation
LatexCommand cite
key "Hall00book"

\end_inset

 by realizing 
\begin_inset Formula $\skew 1$
\end_inset

 is diagonizable with eigenvalues 
\begin_inset Formula $-i$
\end_inset

 and 
\begin_inset Formula $i$
\end_inset

 , and eigenvectors 
\begin_inset Formula $\left[\begin{array}{c}
1\\
i\end{array}\right]$
\end_inset

 and 
\begin_inset Formula $\left[\begin{array}{c}
i\\
1\end{array}\right]$
\end_inset

.
 Readers familiar with projective geometry will recognize these as the circular
 points when promoted to homogeneous coordinates.
\end_layout

\begin_layout Subsection
Adjoint
\end_layout

\begin_layout Standard
The adjoint map for 
\begin_inset Formula $\sotwo$
\end_inset

 is trivially equal to the identity, as is the case for 
\emph on
all
\emph default
 commutative groups:
\begin_inset Formula \begin{eqnarray*}
\Ad R\what & = & \left[\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\end{array}\right]\left[\begin{array}{cc}
0 & -\omega\\
\omega & 0\end{array}\right]\left[\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta\end{array}\right]^{T}\\
 & = & \omega\left[\begin{array}{cc}
-\sin\theta & -\cos\theta\\
\cos\theta & -\sin\theta\end{array}\right]\left[\begin{array}{cc}
\cos\theta & \sin\theta\\
-\sin\theta & \cos\theta\end{array}\right]=\left[\begin{array}{cc}
0 & -\omega\\
\omega & 0\end{array}\right]\end{eqnarray*}

\end_inset

i.e., 
\begin_inset Formula \[
\Ad R\what=\what\]

\end_inset


\end_layout

\begin_layout Subsection
Actions
\end_layout

\begin_layout Standard
In the case of 
\begin_inset Formula $\SOtwo$
\end_inset

 the vector space is 
\begin_inset Formula $\Rtwo$
\end_inset

, and the group action corresponds to rotating a point
\begin_inset Formula \[
q=Rp\]

\end_inset

We would now like to know what an incremental rotation parameterized by
 
\begin_inset Formula $\theta$
\end_inset

 would do:
\begin_inset Formula \[
q(\text{\omega t})=Re^{\skew{\omega t}}p\]

\end_inset

hence the derivative is:
\begin_inset Formula \[
\deriv{q(\omega t)}t=R\deriv{}t\left(e^{\skew{\omega t}}p\right)=R\deriv{}t\left(\skew{\omega t}p\right)=RH_{p}\]

\end_inset

Note that 
\begin_inset Formula \begin{equation}
\skew{\theta}\left[\begin{array}{c}
x\\
y\end{array}\right]=\theta R_{\pi/2}\left[\begin{array}{c}
x\\
y\end{array}\right]=\theta\left[\begin{array}{c}
-y\\
x\end{array}\right]\label{eq:RestrictedCross}\end{equation}

\end_inset

which acts like a restricted 
\begin_inset Quotes eld
\end_inset

cross product
\begin_inset Quotes erd
\end_inset

 in the plane.
\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
2D Rigid Transformations
\end_layout

\begin_layout Subsection
Basics
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $\SEtwo$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(3)$
\end_inset

 of 
\begin_inset Formula $3\times3$
\end_inset

 invertible matrices of the form
\begin_inset Formula \[
T\define\left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\]

\end_inset

where 
\begin_inset Formula $R\in\SOtwo$
\end_inset

 is a rotation matrix and 
\begin_inset Formula $t\in\Rtwo$
\end_inset

 is a translation vector.
 
\begin_inset Formula $\SEtwo$
\end_inset

 is the 
\emph on
semi-direct product
\emph default
 of 
\begin_inset Formula $\Rtwo$
\end_inset

 by 
\begin_inset Formula $SO(2)$
\end_inset

, written as 
\begin_inset Formula $\SEtwo=\Rtwo\rtimes\SOtwo$
\end_inset

.
 In particular, any element 
\begin_inset Formula $T$
\end_inset

 of 
\begin_inset Formula $\SEtwo$
\end_inset

 can be written as
\begin_inset Formula \[
T=\left[\begin{array}{cc}
0 & t\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
R & 0\\
0 & 1\end{array}\right]\]

\end_inset

and they compose as
\begin_inset Formula \[
T_{1}T_{2}=\left[\begin{array}{cc}
R_{1} & t_{1}\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
R_{2} & t_{2}\\
0 & 1\end{array}\right]=\left[\begin{array}{cc}
R_{1}R_{2} & R_{1}t_{2}+t_{1}\\
0 & 1\end{array}\right]\]

\end_inset

Hence, an alternative way of writing down elements of 
\begin_inset Formula $\SEtwo$
\end_inset

 is as the ordered pair 
\begin_inset Formula $(R,\, t)$
\end_inset

, with composition defined a
\begin_inset Formula \[
(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})\]

\end_inset


\end_layout

\begin_layout Standard
The corresponding Lie algebra 
\begin_inset Formula $\setwo$
\end_inset

 is the vector space of 
\begin_inset Formula $3\times3$
\end_inset

 twists 
\begin_inset Formula $\xihat$
\end_inset

 parameterized by the 
\emph on
twist coordinates
\emph default
 
\begin_inset Formula $\xi\in\Rthree$
\end_inset

, with the mapping 
\begin_inset Formula \[
\xi\define\left[\begin{array}{c}
v\\
\omega\end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc}
\skew{\omega} & v\\
0 & 0\end{array}\right]\]

\end_inset

Note we think of robots as having a pose 
\begin_inset Formula $(x,y,\theta)$
\end_inset

 and hence I reserved the first two components for translation and the last
 for rotation.
 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
The corresponding Lie group generators are
\begin_inset Formula \[
G^{x}=\left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{y}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{\theta}=\left[\begin{array}{ccc}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{array}\right]\]

\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
Applying the exponential map to a twist 
\begin_inset Formula $\xi$
\end_inset

 yields a screw motion yielding an element in 
\begin_inset Formula $\SEtwo$
\end_inset

: 
\begin_inset Formula \[
T=e^{\xihat}=\left(e^{\skew{\omega}},(I-e^{\skew{\omega}})\frac{v^{\perp}}{\omega}\right)\]

\end_inset


\end_layout

\begin_layout Subsection
The Adjoint Map
\end_layout

\begin_layout Standard
The adjoint is 
\begin_inset Formula \begin{eqnarray}
\Ad T{\xihat} & = & T\xihat T^{-1}\nonumber \\
 & = & \left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
\skew{\omega} & v\\
0 & 0\end{array}\right]\left[\begin{array}{cc}
R^{T} & -R^{T}t\\
0 & 1\end{array}\right]\nonumber \\
 & = & \left[\begin{array}{cc}
\skew{\omega} & -\skew{\omega}t+Rv\\
0 & 0\end{array}\right]\nonumber \\
 & = & \left[\begin{array}{cc}
\skew{\omega} & Rv-\omega R_{\pi/2}t\\
0 & 0\end{array}\right]\label{eq:adjointSE2}\end{eqnarray}

\end_inset

From this we can express the Adjoint map in terms of plane twist coordinates:
\begin_inset Formula \[
\left[\begin{array}{c}
v'\\
\omega'\end{array}\right]=\left[\begin{array}{cc}
R & -R_{\pi/2}t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
v\\
\omega\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Subsection
Actions
\end_layout

\begin_layout Standard
The action of 
\begin_inset Formula $\SEtwo$
\end_inset

 on 2D points is done by embedding the points in 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 by using homogeneous coordinates
\begin_inset Formula \[
\hat{q}=\left[\begin{array}{c}
q\\
1\end{array}\right]=\left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=T\hat{p}\]

\end_inset

Analoguous to 
\begin_inset Formula $\SEthree$
\end_inset

, we can compute a velocity 
\begin_inset Formula $\xihat\hat{p}$
\end_inset

 in the local 
\begin_inset Formula $T$
\end_inset

 frame: 
\begin_inset Formula \[
\xihat\hat{p}=\left[\begin{array}{cc}
\skew{\omega} & v\\
0 & 0\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=\left[\begin{array}{c}
\skew{\omega}p+v\\
0\end{array}\right]\]

\end_inset

By only taking the top two rows, we can write this as a velocity in 
\begin_inset Formula $\Rtwo$
\end_inset

, as the product of a 
\begin_inset Formula $2\times3$
\end_inset

 matrix 
\begin_inset Formula $H_{p}$
\end_inset

 that acts upon the exponential coordinates 
\begin_inset Formula $\xi$
\end_inset

 directly:
\begin_inset Formula \[
\skew{\omega}p+v=v+R_{\pi/2}p\omega=\left[\begin{array}{cc}
I_{2} & R_{\pi/2}p\end{array}\right]\left[\begin{array}{c}
v\\
\omega\end{array}\right]=H_{p}\xi\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
3D Rotations
\end_layout

\begin_layout Subsection
Basics
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $\SOthree$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(3)$
\end_inset

 of 
\begin_inset Formula $3\times3$
\end_inset

 invertible matrices.
 Its Lie algebra 
\begin_inset Formula $\sothree$
\end_inset

 is the vector space of 
\begin_inset Formula $3\times3$
\end_inset

 skew-symmetric matrices 
\begin_inset Formula $\what$
\end_inset

.
 The exponential map can be computed in closed form using Rodrigues' formula
 
\begin_inset CommandInset citation
LatexCommand cite
after "page 28"
key "Murray94book"

\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula \[
e^{\what}=I+\what\sin\theta+\what^{2}(1\text{−}\cos\theta)\]

\end_inset

where 
\begin_inset Formula $\what^{2}=\omega\omega^{T}-I$
\end_inset

, with 
\begin_inset Formula $\omega\omega^{T}$
\end_inset

 the outer product of 
\begin_inset Formula $\omega$
\end_inset

.
 Hence, a slightly more efficient variant is
\begin_inset Formula \[
e^{\what}=\cos\theta I+\what sin\theta+\omega\omega^{T}(1\text{−}cos\theta)\]

\end_inset

Since 
\begin_inset Formula $\SOthree$
\end_inset

 is a three-dimensional manifold, 
\begin_inset Formula $\sothree$
\end_inset

 is isomorphic to 
\begin_inset Formula $\Rthree$
\end_inset

 and we define the map
\begin_inset Formula \[
\hat{}:\Rthree\rightarrow\sothree\]

\end_inset


\begin_inset Formula \[
\hat{}:\omega\rightarrow\what=\Skew{\omega}\]

\end_inset

which maps 3-vectors 
\begin_inset Formula $\omega$
\end_inset

 to skew-symmetric matrices 
\begin_inset Formula $\Skew{\omega}$
\end_inset

 :
\begin_inset Formula \[
\Skew{\omega}=\left[\begin{array}{ccc}
0 & -\omega_{z} & \omega_{y}\\
\omega_{z} & 0 & -\omega_{x}\\
-\omega_{y} & \omega_{x} & 0\end{array}\right]=\omega_{x}G^{x}+\omega_{y}G^{y}+\omega_{z}G^{z}\]

\end_inset

where the 
\begin_inset Formula $G^{i}$
\end_inset

 are the generators for 
\begin_inset Formula $\SOthree$
\end_inset

,
\begin_inset Formula \[
G^{x}=\left(\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0\end{array}\right)\mbox{}G^{y}=\left(\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
-1 & 0 & 0\end{array}\right)\mbox{ }G^{z}=\left(\begin{array}{ccc}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{array}\right)\]

\end_inset

corresponding to a rotation around 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $Y$
\end_inset

, and 
\begin_inset Formula $Z$
\end_inset

, respectively.
 The Lie bracket 
\begin_inset Formula $[x,y]$
\end_inset

 corresponds to the cross product 
\begin_inset Formula $x\times y$
\end_inset

 in 
\begin_inset Formula $\Rthree$
\end_inset

.
\end_layout

\begin_layout Standard
For every 
\begin_inset Formula $3-$
\end_inset

vector 
\begin_inset Formula $\omega$
\end_inset

 there is a corresponding rotation matrix
\begin_inset Formula \[
R=e^{\Skew{\omega}}\]

\end_inset

and this is defines the canonical parameterization of 
\begin_inset Formula $\SOthree$
\end_inset

, with 
\begin_inset Formula $\omega$
\end_inset

 known as the canonical or exponential coordinates.
 It is equivalent to the axis-angle representation for rotations, where
 the unit vector 
\begin_inset Formula $\omega/\left\Vert \omega\right\Vert $
\end_inset

 defines the rotation axis, and its magnitude the amount of rotation 
\begin_inset Formula $\theta$
\end_inset

.
\end_layout

\begin_layout Subsection
The Adjoint Map
\end_layout

\begin_layout Standard
For rotation matrices 
\begin_inset Formula $R$
\end_inset

 we can prove the following identity (see 
\begin_inset CommandInset ref
LatexCommand vref
reference "proof1"

\end_inset

): 
\begin_inset Formula \begin{equation}
R\Skew{\omega}R^{T}=\Skew{R\omega}\label{eq:property1}\end{equation}

\end_inset

Hence, given property 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:property1"

\end_inset

, the adjoint map for 
\begin_inset Formula $\sothree$
\end_inset

 simplifies to
\begin_inset Formula \[
\Ad R{\Skew{\omega}}=R\Skew{\omega}R^{T}=\Skew{R\omega}\]

\end_inset

and this can be expressed in exponential coordinates simply by rotating
 the axis 
\begin_inset Formula $\omega$
\end_inset

 to 
\begin_inset Formula $R\omega$
\end_inset

.
 
\end_layout

\begin_layout Standard
As an example, to apply an axis-angle rotation 
\begin_inset Formula $\omega$
\end_inset

 to a point 
\begin_inset Formula $p$
\end_inset

 in the frame 
\begin_inset Formula $R$
\end_inset

, we could:
\end_layout

\begin_layout Enumerate
First transform 
\begin_inset Formula $p$
\end_inset

 back to the world frame, apply 
\begin_inset Formula $\omega$
\end_inset

, and then rotate back:
\begin_inset Formula \[
q=Re^{\Skew{\omega}}R^{T}\]

\end_inset


\end_layout

\begin_layout Enumerate
Immediately apply the transformed axis-angle transformation 
\begin_inset Formula $\Ad R{\Skew{\omega}}=\Skew{R\omega}$
\end_inset

:
\begin_inset Formula \[
q=e^{\Skew{R\omega}}p\]

\end_inset


\end_layout

\begin_layout Subsection
Actions
\end_layout

\begin_layout Standard
In the case of 
\begin_inset Formula $\SOthree$
\end_inset

 the vector space is  
\begin_inset Formula $\Rthree$
\end_inset

, and the group action corresponds to rotating a point
\begin_inset Formula \[
q=Rp\]

\end_inset

We would now like to know what an incremental rotation parameterized by
 
\begin_inset Formula $\omega$
\end_inset

 would do:
\begin_inset Formula \[
q(\omega)=Re^{\Skew{\omega}}p\]

\end_inset

hence the derivative is:
\begin_inset Formula \[
\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=R\Skew{-p}\]

\end_inset

To show the last equality note that 
\begin_inset Formula \[
\Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
3D Rigid Transformations
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $\SEthree$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(4)$
\end_inset

 of 
\begin_inset Formula $4\times4$
\end_inset

 invertible matrices of the form
\begin_inset Formula \[
T\define\left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\]

\end_inset

where 
\begin_inset Formula $R\in\SOthree$
\end_inset

 is a rotation matrix and 
\begin_inset Formula $t\in\Rthree$
\end_inset

 is a translation vector.
 An alternative way of writing down elements of 
\begin_inset Formula $\SEthree$
\end_inset

 is as the ordered pair 
\begin_inset Formula $(R,\, t)$
\end_inset

, with composition defined as
\begin_inset Formula \[
(R_{1},\, t_{1})(R_{2},\, t_{2})=(R_{1}R_{2},\, R{}_{1}t_{2}+t_{1})\]

\end_inset

 Its Lie algebra 
\begin_inset Formula $\sethree$
\end_inset

 is the vector space of 
\begin_inset Formula $4\times4$
\end_inset

 twists 
\begin_inset Formula $\xihat$
\end_inset

 parameterized by the 
\emph on
twist coordinates
\emph default
 
\begin_inset Formula $\xi\in\Rsix$
\end_inset

, with the mapping 
\begin_inset CommandInset citation
LatexCommand cite
key "Murray94book"

\end_inset

 
\begin_inset Formula \[
\xi\define\left[\begin{array}{c}
\omega\\
v\end{array}\right]\rightarrow\xihat\define\left[\begin{array}{cc}
\Skew{\omega} & v\\
0 & 0\end{array}\right]\]

\end_inset

Note we follow Frank Park's convention and reserve the first three components
 for rotation, and the last three for translation.
 Hence, with this parameterization, the generators for 
\begin_inset Formula $\SEthree$
\end_inset

 are
\begin_inset Formula \[
G^{1}=\left(\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\mbox{}G^{2}=\left(\begin{array}{cccc}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 0\\
-1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\mbox{ }G^{3}=\left(\begin{array}{cccc}
0 & -1 & 0 & 0\\
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\]

\end_inset


\begin_inset Formula \[
G^{4}=\left(\begin{array}{cccc}
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\mbox{}G^{5}=\left(\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\mbox{ }G^{6}=\left(\begin{array}{cccc}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & 0 & 0\end{array}\right)\]

\end_inset

Applying the exponential map to a twist 
\begin_inset Formula $\xi$
\end_inset

 yields a screw motion yielding an element in 
\begin_inset Formula $\SEthree$
\end_inset

: 
\begin_inset Formula \[
T=\exp\xihat\]

\end_inset

A closed form solution for the exponential map is given in 
\begin_inset CommandInset citation
LatexCommand cite
after "page 42"
key "Murray94book"

\end_inset

.
\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula \[
\exp\left(\left[\begin{array}{c}
\omega\\
v\end{array}\right]t\right)=\left[\begin{array}{cc}
e^{\Skew{\omega}t} & (I-e^{\Skew{\omega}t})\left(\omega\times v\right)+\omega\omega^{T}vt\\
0 & 1\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Subsection
The Adjoint Map
\end_layout

\begin_layout Standard
The adjoint is 
\begin_inset Formula \begin{eqnarray*}
\Ad T{\xihat} & = & T\xihat T^{-1}\\
 & = & \left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
\Skew{\omega} & v\\
0 & 0\end{array}\right]\left[\begin{array}{cc}
R^{T} & -R^{T}t\\
0 & 1\end{array}\right]\\
 & = & \left[\begin{array}{cc}
\Skew{R\omega} & -\Skew{R\omega}t+Rv\\
0 & 0\end{array}\right]\\
 & = & \left[\begin{array}{cc}
\Skew{R\omega} & t\times R\omega+Rv\\
0 & 0\end{array}\right]\end{eqnarray*}

\end_inset

From this we can express the Adjoint map in terms of twist coordinates (see
 also 
\begin_inset CommandInset citation
LatexCommand cite
key "Murray94book"

\end_inset

 and FP):
\begin_inset Formula \[
\left[\begin{array}{c}
\omega'\\
v'\end{array}\right]=\left[\begin{array}{cc}
R & 0\\
\Skew tR & R\end{array}\right]\left[\begin{array}{c}
\omega\\
v\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Subsection
Actions
\end_layout

\begin_layout Standard
The action of 
\begin_inset Formula $\SEthree$
\end_inset

 on 3D points is done by embedding the points in 
\begin_inset Formula $\mathbb{R}^{4}$
\end_inset

 by using homogeneous coordinates
\begin_inset Formula \[
\hat{q}=\left[\begin{array}{c}
q\\
1\end{array}\right]=\left[\begin{array}{c}
Rp+t\\
1\end{array}\right]=\left[\begin{array}{cc}
R & t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=T\hat{p}\]

\end_inset

We would now like to know what an incremental rotation parameterized by
 
\begin_inset Formula $\xi$
\end_inset

 would do:
\begin_inset Formula \[
\hat{q}(\xi)=Te^{\xihat}\hat{p}\]

\end_inset

hence the derivative is
\begin_inset Formula \[
\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)\]

\end_inset

where 
\begin_inset Formula $\xihat\hat{p}$
\end_inset

 corresponds to a velocity in 
\begin_inset Formula $\mathbb{R}^{4}$
\end_inset

 (in the local 
\begin_inset Formula $T$
\end_inset

 frame): 
\begin_inset Formula \[
\xihat\hat{p}=\left[\begin{array}{cc}
\Skew{\omega} & v\\
0 & 0\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=\left[\begin{array}{c}
\omega\times p+v\\
0\end{array}\right]\]

\end_inset

Notice how velocities are anologous to points at infinity in projective
 geometry: they correspond to free vectors indicating a direction and magnitude
 of change.
\end_layout

\begin_layout Standard
By only taking the top three rows, we can write this as a velocity in 
\begin_inset Formula $\Rthree$
\end_inset

, as the product of a 
\begin_inset Formula $3\times6$
\end_inset

 matrix 
\begin_inset Formula $H_{p}$
\end_inset

 that acts upon the exponential coordinates 
\begin_inset Formula $\xi$
\end_inset

 directly:
\begin_inset Formula \[
\omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc}
-\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c}
\omega\\
v\end{array}\right]\]

\end_inset

The inverse action 
\begin_inset Formula $T^{-1}p$
\end_inset

 is
\begin_inset Formula \[
\hat{q}=\left[\begin{array}{c}
q\\
1\end{array}\right]=\left[\begin{array}{c}
R^{T}(p-t)\\
1\end{array}\right]=\left[\begin{array}{cc}
R^{T} & -R^{T}t\\
0 & 1\end{array}\right]\left[\begin{array}{c}
p\\
1\end{array}\right]=T^{-1}\hat{p}\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section
2D Affine Transformations
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $Aff(2)$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(3)$
\end_inset

 of 
\begin_inset Formula $3\times3$
\end_inset

 invertible matrices that maps the line infinity to itself, and hence preserves
 paralellism.
 The affine transformation matrices 
\begin_inset Formula $A$
\end_inset

 can be written as 
\begin_inset CommandInset citation
LatexCommand cite
key "Mei08tro"

\end_inset


\family roman
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\color none

\begin_inset Formula \[
\left[\begin{array}{ccc}
m_{11} & m_{12} & t_{1}\\
m_{21} & m_{22} & t_{2}\\
0 & 0 & k\end{array}\right]\]

\end_inset

with 
\begin_inset Formula $M\in GL(2)$
\end_inset

, 
\begin_inset Formula $t\in\Rtwo$
\end_inset

, and 
\begin_inset Formula $k$
\end_inset

 a scalar chosen such that 
\begin_inset Formula $det(A)=1$
\end_inset

.
 
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Note that just as 
\begin_inset Formula $\SEtwo$
\end_inset

 is a semi-direct product, so too is 
\begin_inset Formula $Aff(2)=\Rtwo\rtimes GL(2)$
\end_inset

.
 In particular, any affine transformation 
\begin_inset Formula $A$
\end_inset

 can be written as
\begin_inset Formula \[
A=\left[\begin{array}{cc}
0 & t\\
0 & 1\end{array}\right]\left[\begin{array}{cc}
M & 0\\
0 & k\end{array}\right]\]

\end_inset

and they compose as
\begin_inset Formula \[
A_{1}A_{2}=\left[\begin{array}{cc}
M_{1} & t_{1}\\
0 & k_{1}\end{array}\right]\left[\begin{array}{cc}
M_{2} & t_{2}\\
0 & k_{2}\end{array}\right]=\left[\begin{array}{cc}
M_{1}M_{2} & M_{2}t_{2}+k_{2}t_{1}\\
0 & k_{1}k_{2}\end{array}\right]\]

\end_inset

From this it can be gleaned that the groups 
\begin_inset Formula $\SOtwo$
\end_inset

 and 
\begin_inset Formula $\SEtwo$
\end_inset

 are both subgroups, with 
\begin_inset Formula $\SOtwo\subset\SEtwo\subset\Afftwo$
\end_inset

.
 
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By choosing the generators carefully we maintain this hierarchy among the
 associated Lie algebras.
 In particular, 
\begin_inset Formula $\setwo$
\end_inset

 
\begin_inset Formula \[
G^{1}=\left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{2}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{3}=\left[\begin{array}{ccc}
0 & -1 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{array}\right]\]

\end_inset

can be extended to the 
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Lie algebra
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\begin_inset Formula $\afftwo$
\end_inset

 using the three additional generators
\begin_inset Formula \[
G^{4}=\left[\begin{array}{ccc}
0 & 1 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{5}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 1\end{array}\right]\]

\end_inset


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Hence, the Lie algebra 
\begin_inset Formula $\afftwo$
\end_inset

 is the vector space of 
\begin_inset Formula $3\times3$
\end_inset

 incremental affine transformations 
\begin_inset Formula $\ahat$
\end_inset

 parameterized by 6 parameters 
\begin_inset Formula $\aa\in\mathbb{R}^{6}$
\end_inset

, with the mapping 
\begin_inset Formula \[
\aa\rightarrow\ahat\define\left[\begin{array}{ccc}
a_{5} & a_{4}-a_{3} & a_{1}\\
a_{4}+a_{3} & -a_{5}-a_{6} & a_{2}\\
0 & 0 & a_{6}\end{array}\right]\]

\end_inset

Note that 
\begin_inset Formula $G_{5}$
\end_inset

 and 
\begin_inset Formula $G_{6}$
\end_inset

 change the relative scale of 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 but without changing the determinant: 
\begin_inset Formula \[
e^{xG_{5}}=\exp\left[\begin{array}{ccc}
x & 0 & 0\\
0 & -x & 0\\
0 & 0 & 0\end{array}\right]=\left[\begin{array}{ccc}
e^{x} & 0 & 0\\
0 & 1/e^{x} & 0\\
0 & 0 & 1\end{array}\right]\]

\end_inset


\begin_inset Formula \[
e^{xG_{6}}=\exp\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & -x & 0\\
0 & 0 & x\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 1/e^{x} & 0\\
0 & 0 & e^{x}\end{array}\right]\]

\end_inset

It might be nicer to have the correspondence with scaling 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $y$
\end_inset

 more direct, by choosing
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\begin_inset Formula \[
\mbox{ }G^{5}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -1\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1\end{array}\right]\]

\end_inset

and hence
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\begin_inset Formula \[
e^{xG_{5}}=\exp\left[\begin{array}{ccc}
x & 0 & 0\\
0 & 0 & 0\\
0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc}
e^{x} & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1/e^{x}\end{array}\right]\]

\end_inset


\begin_inset Formula \[
e^{xG_{6}}=\exp\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & x & 0\\
0 & 0 & -x\end{array}\right]=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & e^{x} & 0\\
0 & 0 & 1/e^{x}\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Section
2D Homographies
\end_layout

\begin_layout Standard
When viewed as operations on images, represented by 2D projective space
 
\begin_inset Formula $\mathcal{P}^{3}$
\end_inset

, 3D rotations are a special case of 2D homographies.
 These are now treated, loosely based on the exposition in 
\begin_inset CommandInset citation
LatexCommand cite
key "Mei06iros,Mei08tro"

\end_inset

.
\end_layout

\begin_layout Subsection
Basics
\end_layout

\begin_layout Standard
The Lie group 
\begin_inset Formula $\SLthree$
\end_inset

 is a subgroup of the general linear group 
\begin_inset Formula $GL(3)$
\end_inset

 of 
\begin_inset Formula $3\times3$
\end_inset

 invertible matrices with determinant 
\begin_inset Formula $1$
\end_inset

.
 The homographies generalize transformations of the 2D projective space,
 and 
\begin_inset Formula $\Afftwo\subset\SLthree$
\end_inset

.
 
\end_layout

\begin_layout Standard

\family roman
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\shape up
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\emph off
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\noun off
\color none
We can extend 
\begin_inset Formula $\afftwo$
\end_inset

 to the Lie algebra 
\begin_inset Formula $\slthree$
\end_inset

 by adding two generators
\begin_inset Formula \[
G^{7}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
1 & 0 & 0\end{array}\right]\mbox{ }G^{8}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 1 & 0\end{array}\right]\]

\end_inset


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obtaining the vector space of 
\begin_inset Formula $3\times3$
\end_inset

 incremental homographies 
\begin_inset Formula $\hhat$
\end_inset

 parameterized by 8 parameters 
\begin_inset Formula $\hh\in\mathbb{R}^{8}$
\end_inset

, with the mapping 
\begin_inset Formula \[
h\rightarrow\hhat\define\left[\begin{array}{ccc}
h_{5} & h_{4}-h_{3} & h_{1}\\
h_{4}+h_{3} & -h_{5}-h_{6} & h_{2}\\
h_{7} & h_{8} & h_{6}\end{array}\right]\]

\end_inset


\end_layout

\begin_layout Subsection
Tensor Notation
\end_layout

\begin_layout Itemize
A homography between 2D projective spaces 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

 can be written in tensor notation 
\begin_inset Formula $H_{A}^{B}$
\end_inset


\end_layout

\begin_layout Itemize
Applying a homography is then a tensor contraction 
\begin_inset Formula $x^{B}=H_{A}^{B}x^{A}$
\end_inset

, mapping points in 
\begin_inset Formula $A$
\end_inset

 to points in 
\begin_inset Formula $B$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
The inverse of a homography can be found by contracting with two permutation
 tensors:
\begin_inset Formula \[
H_{B}^{A}=H_{A_{1}}^{B_{1}}H_{A_{2}}^{B_{2}}\epsilon_{B_{1}B_{2}B}\epsilon^{A_{1}A_{2}A}\]

\end_inset


\end_layout

\end_inset


\begin_inset Note Note
status collapsed

\begin_layout Subsection
The Adjoint Map
\end_layout

\begin_layout Plain Layout
The adjoint can be done using tensor notation.
 Denoting an incremental homography in space 
\begin_inset Formula $A$
\end_inset

 as 
\begin_inset Formula $\hhat_{A_{1}}^{A_{2}}$
\end_inset

, we have, for example for 
\begin_inset Formula $G_{1}$
\end_inset


\begin_inset Formula \begin{eqnarray*}
\hhat_{B_{1}}^{B_{2}}=\Ad{H_{A}^{B}}{\hhat_{A_{1}}^{A_{2}}} & = & H_{A_{2}}^{B_{2}}\hhat_{A_{1}}^{A_{2}}H_{B_{1}}^{A_{1}}\\
 & = & H_{A_{2}}^{B_{2}}\left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right]H_{A_{2}}^{B_{2}}H_{A_{3}}^{B_{3}}\epsilon_{B_{1}B_{2}B_{3}}\epsilon^{A_{1}A_{2}A_{3}}\\
 & = & H_{1}^{B_{2}}H_{A_{2}}^{B_{2}}H_{A_{3}}^{B_{3}}\epsilon_{B_{1}B_{2}B_{3}}\epsilon^{3A_{2}A_{3}}\end{eqnarray*}

\end_inset

This does not seem to help.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage pagebreak
\end_inset


\end_layout

\begin_layout Section*
Appendix: Proof of Property 
\begin_inset CommandInset ref
LatexCommand ref
reference "proof1"

\end_inset


\end_layout

\begin_layout Standard
We can prove the following identity for rotation matrices 
\begin_inset Formula $R$
\end_inset

,
\begin_inset Formula \begin{eqnarray}
R\Skew{\omega}R^{T} & = & R\Skew{\omega}\left[\begin{array}{ccc}
a_{1} & a_{2} & a_{3}\end{array}\right]\nonumber \\
 & = & R\left[\begin{array}{ccc}
\omega\times a_{1} & \omega\times a_{2} & \omega\times a_{3}\end{array}\right]\nonumber \\
 & = & \left[\begin{array}{ccc}
a_{1}(\omega\times a_{1}) & a_{1}(\omega\times a_{2}) & a_{1}(\omega\times a_{3})\\
a_{2}(\omega\times a_{1}) & a_{2}(\omega\times a_{2}) & a_{2}(\omega\times a_{3})\\
a_{3}(\omega\times a_{1}) & a_{3}(\omega\times a_{2}) & a_{3}(\omega\times a_{3})\end{array}\right]\nonumber \\
 & = & \left[\begin{array}{ccc}
\omega(a_{1}\times a_{1}) & \omega(a_{2}\times a_{1}) & \omega(a_{3}\times a_{1})\\
\omega(a_{1}\times a_{2}) & \omega(a_{2}\times a_{2}) & \omega(a_{3}\times a_{2})\\
\omega(a_{1}\times a_{3}) & \omega(a_{2}\times a_{3}) & \omega(a_{3}\times a_{3})\end{array}\right]\nonumber \\
 & = & \left[\begin{array}{ccc}
0 & -\omega a_{3} & \omega a_{2}\\
\omega a_{3} & 0 & -\omega a_{1}\\
-\omega a_{2} & \omega a_{1} & 0\end{array}\right]\nonumber \\
 & = & \Skew{R\omega}\label{proof1}\end{eqnarray}

\end_inset

where 
\begin_inset Formula $a_{1}$
\end_inset

, 
\begin_inset Formula $a_{2}$
\end_inset

, and 
\begin_inset Formula $a_{3}$
\end_inset

 are the 
\emph on
rows
\emph default
 of 
\begin_inset Formula $R$
\end_inset

.
 Above we made use of the orthogonality of rotation matrices and the triple
 product rule:
\begin_inset Formula \[
a(b\times c)=b(c\times a)=c(a\times b)\]

\end_inset

Similarly, without proof 
\begin_inset CommandInset citation
LatexCommand cite
after "Lemma 2.3"
key "Murray94book"

\end_inset

: 
\begin_inset Formula \[
R(a\times b)=Ra\times Rb\]

\end_inset


\end_layout

\begin_layout Section*
Appendix: Alternative Generators for 
\begin_inset Formula $\slthree$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset citation
LatexCommand cite
key "Mei06iros"

\end_inset

 uses the following generators for 
\begin_inset Formula $\slthree$
\end_inset

:
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\color none

\begin_inset Formula \[
G^{1}=\left[\begin{array}{ccc}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{2}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 1\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{3}=\left[\begin{array}{ccc}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 0\end{array}\right]\]

\end_inset


\begin_inset Formula \[
G^{4}=\left[\begin{array}{ccc}
0 & 0 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{5}=\left[\begin{array}{ccc}
1 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 0\end{array}\right]\mbox{ }G^{6}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & -1 & 0\\
0 & 0 & 1\end{array}\right]\]

\end_inset


\begin_inset Formula \[
G^{7}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
1 & 0 & 0\end{array}\right]\mbox{ }G^{8}=\left[\begin{array}{ccc}
0 & 0 & 0\\
0 & 0 & 0\\
0 & 1 & 0\end{array}\right]\]

\end_inset


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We choose to use a different linear combination as the basis.
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "../../../papers/refs"
options "plain"

\end_inset


\end_layout

\end_body
\end_document