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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
-\begin_inset Note Comment
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-Derivatives
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-\begin_inset Note Comment
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-\begin_layout Plain Layout
-Lie Groups
-\end_layout
-
-\end_inset
-
-
-\end_layout
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-\begin_inset FormulaMacro
-\newcommand{\gg}{\mathfrak{g}}
-{\mathfrak{g}}
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-
-\begin_inset FormulaMacro
-\newcommand{\Rn}{\mathbb{R}^{n}}
-{\mathbb{R}^{n}}
-\end_inset
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-\begin_layout Standard
-\begin_inset Note Comment
-status open
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-\begin_layout Plain Layout
-SO(2), 1
-\end_layout
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-
-\end_layout
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-\begin_layout Standard
-\begin_inset FormulaMacro
-\newcommand{\Rtwo}{\mathfrak{\mathbb{R}^{2}}}
-{\mathfrak{\mathbb{R}^{2}}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\SOtwo}{SO(2)}
-{SO(2)}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\sotwo}{\mathfrak{so(2)}}
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-SE(2), 3
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-\newcommand{\SEtwo}{SE(2)}
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-SO(3), 3
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-SL(3),8
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-\begin_layout Standard
-\begin_inset FormulaMacro
-\newcommand{\SLthree}{SL(3)}
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-\newcommand{\hhat}{\hat{h}}
-{\hat{h}}
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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
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\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
-
-.
- 
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
-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
-
-.
- 
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-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 
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
-Lie algebra
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
- 
-\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
-
-
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
-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
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-
-\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
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
- 
-\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
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\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
-
-
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
-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}
-
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-	<key>OutlineStyle</key>
-	<string>Basic</string>
-	<key>PageBreaks</key>
-	<string>NO</string>
-	<key>PrintInfo</key>
-	<dict>
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-		<array>
-			<string>float</string>
-			<string>41</string>
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-		<key>NSLeftMargin</key>
-		<array>
-			<string>float</string>
-			<string>18</string>
-		</array>
-		<key>NSPaperSize</key>
-		<array>
-			<string>size</string>
-			<string>{595.29, 841.89}</string>
-		</array>
-		<key>NSRightMargin</key>
-		<array>
-			<string>float</string>
-			<string>18</string>
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-		<key>NSTopMargin</key>
-		<array>
-			<string>float</string>
-			<string>18</string>
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-	<key>PrintOnePage</key>
-	<true/>
-	<key>QuickLookPreview</key>
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-\newcommand{\setwo}{\mathfrak{se(2)}}
-{\mathfrak{se(2)}}
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Note Comment
-status open
-
-\begin_layout Plain Layout
-SO(3)
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset FormulaMacro
-\newcommand{\Rthree}{\mathfrak{\mathbb{R}^{3}}}
-{\mathfrak{\mathbb{R}^{3}}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\SOthree}{SO(3)}
-{SO(3)}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\sothree}{\mathfrak{so(3)}}
-{\mathfrak{so(3)}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\what}{\hat{\omega}}
-{\hat{\omega}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\Skew}[1]{[#1]_{\times}}
-{[#1]_{\times}}
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset Note Comment
-status open
-
-\begin_layout Plain Layout
-SE(3)
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset FormulaMacro
-\newcommand{\Rsix}{\mathfrak{\mathbb{R}^{6}}}
-{\mathfrak{\mathbb{R}^{6}}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\SEthree}{SE(3)}
-{SE(3)}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\sethree}{\mathfrak{se(3)}}
-{\mathfrak{se(3)}}
-\end_inset
-
-
-\begin_inset FormulaMacro
-\newcommand{\xihat}{\hat{\xi}}
-{\hat{\xi}}
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Derivatives of Lie Group Mappings
-\end_layout
-
-\begin_layout Subsection
-Homomorphisms
-\end_layout
-
-\begin_layout Standard
-The following is relevant 
-\begin_inset CommandInset citation
-LatexCommand cite
-after "page 45"
-key "Hall00book"
-
-\end_inset
-
-: suppose that 
-\begin_inset Formula $\Phi:G\rightarrow H$
-\end_inset
-
- is a a mapping (Lie group homomorphism).
- Then there exists a unique linear map 
-\begin_inset Formula $\phi:\gg\rightarrow\mathfrak{h}$
-\end_inset
-
- 
-\begin_inset Formula \[
-\phi(\xhat)\define\lim_{t\rightarrow0}\frac{d}{dt}\Phi\left(e^{t\xhat}\right)\]
-
-\end_inset
-
-such that
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\Phi\left(e^{\xhat}\right)=e^{\phi\left(\xhat\right)}$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\phi\left(T\xhat T^{-1}\right)=\Phi(T)\phi(\xhat)\Phi(T^{-1})$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\phi\left([\xhat,\yhat]\right)=\left[\phi(\xhat),\phi(\yhat)\right]$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In other words, the map 
-\begin_inset Formula $\phi$
-\end_inset
-
- is the derivative of 
-\begin_inset Formula $\Phi$
-\end_inset
-
- at the identity.
- It suffices to compute 
-\begin_inset Formula $\phi$
-\end_inset
-
- for a basis of 
-\begin_inset Formula $\gg$
-\end_inset
-
-.
- Since 
-\begin_inset Formula \[
-e^{-\xhat}=\left(e^{-\xhat}\right)^{-1}\]
-
-\end_inset
-
- clearly 
-\begin_inset Formula $\phi(\xhat)=-\xhat$
-\end_inset
-
- for the inverse mapping.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Plain Layout
-Let us define two mappings
-\begin_inset Formula \[
-\Phi_{1}(A)=AB\mbox{ and }\Phi_{2}(B)=AB\]
-
-\end_inset
-
-Then 
-\begin_inset Formula \[
-\phi_{1}(\xhat)=\lim_{t\rightarrow0}\frac{d}{dt}\Phi_{1}\left(e^{t\xhat}B\right)=\]
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Derivatives
-\end_layout
-
-\begin_layout Standard
-The derivatives for 
-\emph on
-inverse, compose
-\emph default
-, and 
-\emph on
-between
-\emph default
- can be derived from Lie group principles.
- 
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Plain Layout
-To find the derivatives of these functions, we look for the necessary 
-\begin_inset Quotes eld
-\end_inset
-
-delta
-\begin_inset Quotes erd
-\end_inset
-
- in the tangent space of the function 
-\emph on
-output
-\emph default
- 
-\begin_inset Formula $f(g)$
-\end_inset
-
- that corresponds to a 
-\begin_inset Quotes eld
-\end_inset
-
-delta
-\begin_inset Quotes erd
-\end_inset
-
- in the tangent space of the function 
-\emph on
-input
-\emph default
- 
-\begin_inset Formula $g$
-\end_inset
-
-.
- 
-\end_layout
-
-\end_inset
-
-Specifically, to find the derivative of a function 
-\begin_inset Formula $f\left(g\right)$
-\end_inset
-
-, we want to find the Lie algebra element 
-\begin_inset Formula $\yhat\in\gg$
-\end_inset
-
-, that will result from changing 
-\begin_inset Formula $g$
-\end_inset
-
- using 
-\begin_inset Formula $\xhat$
-\end_inset
-
-, also in exponential coordinates:
-\begin_inset Formula \[
-f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\]
-
-\end_inset
-
-Calculating these derivatives requires that we know the form of the function
- 
-\begin_inset Formula $f$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Starting with 
-\series bold
-inverse
-\series default
-, i.e., 
-\begin_inset Formula $f(g)=g^{-1}$
-\end_inset
-
-, we have
-\begin_inset Formula \begin{align}
-g^{-1}e^{\yhat} & =\left(ge^{\xhat}\right)^{-1}=e^{-\xhat}g^{-1}\nonumber \\
-e^{\yhat} & =ge^{-\xhat}g^{-1}=e^{\Ad g\left(-\xhat\right)}\nonumber \\
-\yhat & =\Ad g\left(-\xhat\right)\label{eq:Dinverse}\end{align}
-
-\end_inset
-
-In other words, and this is very intuitive in hindsight, the inverse is
- just negation of 
-\begin_inset Formula $\xhat$
-\end_inset
-
-, along with an adjoint to make sure it is applied in the right frame!
-\end_layout
-
-\begin_layout Standard
-
-\series bold
-Compose
-\series default
- can be derived similarly.
- Let us define two functions to find the derivatives in first and second
- arguments:
-\begin_inset Formula \[
-f_{1}(g)=gh\mbox{ and }f_{2}(h)=gh\]
-
-\end_inset
-
- The latter is easiest, as a change 
-\begin_inset Formula $\xhat$
-\end_inset
-
- in the second argument 
-\begin_inset Formula $h$
-\end_inset
-
- simply gets applied to the result 
-\begin_inset Formula $gh$
-\end_inset
-
-: 
-\begin_inset Formula \begin{align}
-f_{2}(h)e^{\yhat} & =f_{2}\left(he^{\xhat}\right)\nonumber \\
-ghe^{\yhat} & =ghe^{\xhat}\nonumber \\
-\yhat & =\xhat\label{eq:Dcompose2}\end{align}
-
-\end_inset
-
-The derivative for the first argument is a bit trickier:
-\begin_inset Formula \begin{align}
-f_{1}(g)e^{\yhat} & =f_{1}\left(ge^{\xhat}\right)\nonumber \\
-ghe^{\yhat} & =ge^{\xhat}h\nonumber \\
-e^{\yhat} & =h^{-1}e^{\xhat}h=e^{\Ad{h^{-1}}\xhat}\nonumber \\
-\yhat & =\Ad{h^{-1}}\xhat\label{eq:Dcompose1}\end{align}
-
-\end_inset
-
-In other words, to apply a change 
-\begin_inset Formula $\xhat$
-\end_inset
-
- in 
-\begin_inset Formula $g$
-\end_inset
-
- we first need to undo 
-\begin_inset Formula $h$
-\end_inset
-
-, then apply 
-\begin_inset Formula $\xhat$
-\end_inset
-
-, and then apply 
-\begin_inset Formula $h$
-\end_inset
-
- again.
- All can be done in one step by simply applying 
-\begin_inset Formula $\Ad{h^{-1}}\xhat$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Finally, let us find the derivative of 
-\series bold
-between
-\series default
-, defined as 
-\begin_inset Formula $between(g,h)=compose(inverse(g),h)$
-\end_inset
-
-.
- The derivative in the second argument 
-\begin_inset Formula $h$
-\end_inset
-
- is similarly trivial: 
-\begin_inset Formula $\yhat=\xhat$
-\end_inset
-
-.
- The first argument goes as follows:
-\begin_inset Formula \begin{align}
-f_{1}(g)e^{\yhat} & =f_{1}\left(ge^{\xhat}\right)\nonumber \\
-g^{-1}he^{\yhat} & =\left(ge^{\xhat}\right)^{-1}h=e^{\left(-\xhat\right)}g^{-1}h\nonumber \\
-e^{\yhat} & =\left(h^{-1}g\right)e^{\left(-\xhat\right)}\left(h^{-1}g\right)^{-1}=e^{\Ad{\left(h^{-1}g\right)}\left(-\xhat\right)}\nonumber \\
-\yhat & =\Ad{\left(h^{-1}g\right)}\left(-\xhat\right)=\Ad{between\left(h,g\right)}\left(-\xhat\right)\label{eq:Dbetween1}\end{align}
-
-\end_inset
-
-Hence, now we undo 
-\begin_inset Formula $h$
-\end_inset
-
- and then apply the inverse 
-\begin_inset Formula $\left(-\xhat\right)$
-\end_inset
-
- in the 
-\begin_inset Formula $g$
-\end_inset
-
- frame.
-\end_layout
-
-\begin_layout Subsection*
-Numerical Derivatives
-\end_layout
-
-\begin_layout Standard
-Let's examine
-\begin_inset Formula \[
-f\left(g\right)e^{\yhat}=f\left(ge^{\xhat}\right)\]
-
-\end_inset
-
-and multiply with 
-\begin_inset Formula $f(g)^{-1}$
-\end_inset
-
- on both sides then take the log (which in our case returns 
-\begin_inset Formula $y$
-\end_inset
-
-, not 
-\begin_inset Formula $\yhat$
-\end_inset
-
-):
-\begin_inset Formula \[
-y(x)=\log\left[f\left(g\right)^{-1}f\left(ge^{\xhat}\right)\right]\]
-
-\end_inset
-
-Let us look at 
-\begin_inset Formula $x=0$
-\end_inset
-
-, and perturb in direction 
-\begin_inset Formula $i$
-\end_inset
-
-, 
-\begin_inset Formula $e_{i}=[0,0,d,0,0]$
-\end_inset
-
-.
- Then take derivative, 
-\begin_inset Formula \[
-\deriv{y(d)}d\define\lim_{d->0}\frac{y(d)-y(0)}{d}=\lim_{d->0}\frac{1}{d}\log\left[f\left(g\right)^{-1}f\left(ge^{\hat{e_{i}}}\right)\right]\]
-
-\end_inset
-
-which is the basis for a numerical derivative scheme.
-\end_layout
-
-\begin_layout Standard
-Let us also look at a chain rule.
- If we know the behavior at the origin 
-\begin_inset Formula $I$
-\end_inset
-
-, we can extrapolate
-\begin_inset Formula \[
-f(ge^{\xhat})=f(ge^{\xhat}g^{-1}g)=f(e^{\Ad g\xhat}g)\]
-
-\end_inset
-
- 
-\end_layout
-
-\begin_layout Section
-Derivatives of Actions
-\begin_inset CommandInset label
-LatexCommand label
-name "sec:Derivatives-of-Actions"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Forward Action
-\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
-
-.
- Let us first do away with the derivative in 
-\begin_inset Formula $p$
-\end_inset
-
-, which is easy:
-\begin_inset Formula \[
-\deriv{\left(Tp\right)}p=T\]
-
-\end_inset
-
-We would now like to know what an incremental action 
-\begin_inset Formula $\xhat$
-\end_inset
-
- would do, through the exponential map
-\begin_inset Formula \[
-q(x)=Te^{\xhat}p\]
-
-\end_inset
-
-with derivative
-\begin_inset Formula \[
-\deriv{q(x)}x=T\deriv{}x\left(e^{\xhat}p\right)\]
-
-\end_inset
-
-Since the matrix exponential is given by the series
-\begin_inset Formula \[
-e^{A}=I+A+\frac{A^{2}}{2!}+\frac{A^{3}}{3!}+\ldots\]
-
-\end_inset
-
-we have, to first order
-\begin_inset Formula \[
-e^{\xhat}p=p+\xhat p+\ldots\]
-
-\end_inset
-
-and the derivative of an incremental action x for matrix Lie groups becomes
-\begin_inset Formula \[
-\deriv{q(x)}x=T\deriv{\left(\xhat p\right)}x\define TH_{p}\]
-
-\end_inset
-
-where 
-\begin_inset Formula $H_{p}$
-\end_inset
-
- is an 
-\begin_inset Formula $n\times n$
-\end_inset
-
- Jacobian matrix that depends on 
-\begin_inset Formula $p$
-\end_inset
-
-.
- 
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Plain Layout
-Recalling the definition 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:generators"
-
-\end_inset
-
- of the map 
-\begin_inset Formula $x\rightarrow\xhat$
-\end_inset
-
-, we can calculate 
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-
-\begin_inset Formula $\xhat p$
-\end_inset
-
-
-\family default
-\series default
-\shape default
-\size default
-\emph default
-\bar default
-\noun default
-\color inherit
- as (using tensor notation)
-\begin_inset Formula \[
-\left(\xhat p\right)_{jk}=G_{jk}^{i}x_{i}p^{k}\]
-
-\end_inset
-
-and hence the derivative is
-\begin_inset Formula \[
-\left(H_{p}\right)_{j}^{i}=G_{jk}^{i}p^{k}\]
-
-\end_inset
-
-and the final derivative becomes
-\begin_inset Formula \[
-\deriv{q(x)}x=TH_{p}\]
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Inverse Action
-\end_layout
-
-\begin_layout Standard
-When we apply the inverse transformation 
-\begin_inset Formula \[
-q=T^{-1}p\]
-
-\end_inset
-
-we would now like to know what an incremental action 
-\begin_inset Formula $\xhat$
-\end_inset
-
- on 
-\begin_inset Formula $T$
-\end_inset
-
- would do:
-\begin_inset Formula \begin{eqnarray*}
-q(x) & = & \left(Te^{\xhat}\right)^{-1}p\\
- & = & e^{-\xhat}T^{-1}p\\
- & = & e^{-\xhat}q\\
- & \approx & q-\xhat q\end{eqnarray*}
-
-\end_inset
-
-Hence
-\begin_inset Formula \begin{equation}
-\deriv{q(x)}x=\deriv{\left(q-\xhat q\right)}x=-\deriv{\left(\xhat q\right)}x=-H_{q}\label{eq:inverseAction}\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $H_{q}$
-\end_inset
-
- will be as above.
-\end_layout
-
-\begin_layout Section
-Point3
-\end_layout
-
-\begin_layout Standard
-A cross product 
-\begin_inset Formula $a\times b$
-\end_inset
-
- can be written as a matrix multiplication
-\begin_inset Formula \[
-a\times b=\Skew ab\]
-
-\end_inset
-
-where 
-\begin_inset Formula $\Skew a$
-\end_inset
-
- is a skew-symmetric matrix defined as
-\begin_inset Formula \[
-\Skew{x,y,z}=\left[\begin{array}{ccc}
-0 & -z & y\\
-z & 0 & -x\\
--y & x & 0\end{array}\right]\]
-
-\end_inset
-
-We also have
-\begin_inset Formula \[
-a^{T}\Skew b=-(\Skew ba)^{T}=-(a\times b)^{T}\]
-
-\end_inset
-
-The derivative of a cross product 
-\begin_inset Formula \begin{equation}
-\frac{\partial(a\times b)}{\partial a}=\Skew{-b}\label{eq:Dcross1}\end{equation}
-
-\end_inset
-
-
-\begin_inset Formula \begin{equation}
-\frac{\partial(a\times b)}{\partial b}=\Skew a\label{eq:Dcross2}\end{equation}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-2D Rotations
-\end_layout
-
-\begin_layout Subsection
-Rot2 (in gtsam)
-\end_layout
-
-\begin_layout Standard
-A rotation is stored as 
-\begin_inset Formula $(\cos\theta,\sin\theta)$
-\end_inset
-
-.
- An incremental rotation is applied using the trigonometric sum rule:
-\begin_inset Formula \[
-\cos\theta'=\cos\theta\cos\delta-\sin\theta\sin\delta\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-\sin\theta'=\sin\theta\cos\delta+\cos\theta\sin\delta\]
-
-\end_inset
-
-where 
-\begin_inset Formula $\delta$
-\end_inset
-
- is an incremental rotation angle.
-\end_layout
-
-\begin_layout Subsection
-Derivatives of Mappings
-\end_layout
-
-\begin_layout Standard
-We have the derivative of 
-\series bold
-inverse
-\series default
-,
-\begin_inset Formula \[
-\frac{\partial R^{T}}{\partial\theta}=-\Ad R=-1\mbox{ }\]
-
-\end_inset
-
-
-\series bold
-compose,
-\series default
-
-\begin_inset Formula \[
-\frac{\partial\left(R_{1}R_{2}\right)}{\partial\theta_{1}}=\Ad{R_{2}^{T}}=1\mbox{ and }\frac{\partial\left(R_{1}R_{2}\right)}{\partial\theta_{2}}=1\]
-
-\end_inset
-
-
-\series bold
-
-\begin_inset Formula $and$
-\end_inset
-
-between:
-\series default
-
-\begin_inset Formula \[
-\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\theta_{1}}=-\Ad{R_{2}^{T}R_{1}}=-1\mbox{ and }\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\theta_{2}}=1\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Derivatives of 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
-
-The derivative is (following the exposition in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Derivatives-of-Actions"
-
-\end_inset
-
-):
-\begin_inset Formula \[
-\deriv{q(\omega t)}t=R\deriv{}t\left(e^{\skew{\omega t}}p\right)=R\deriv{}t\left(\skew{\omega t}p\right)\]
-
-\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.
- Hence 
-\begin_inset Formula \[
-\skew{\theta}p=\left[\begin{array}{c}
--y\\
-x\end{array}\right]\theta=\omega R_{pi/2}pt\]
-
-\end_inset
-
-Hence, the final derivative of an action in its first argument is
-\begin_inset Formula \[
-\deriv{q(\omega t)}{\omega t}=\omega RR_{pi/2}p=\omega R_{pi/2}Rp=\omega R_{pi/2}q\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Really need to think of relationship 
-\begin_inset Formula $\omega$
-\end_inset
-
- and 
-\begin_inset Formula $t$
-\end_inset
-
-.
- We don't have a time 
-\begin_inset Formula $t$
-\end_inset
-
- in our code.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-2D Rigid Transformations
-\end_layout
-
-\begin_layout Subsection
-Derivatives of Mappings
-\end_layout
-
-\begin_layout Standard
-We can just define all derivatives in terms of the above adjoint map:
-\begin_inset Formula \begin{eqnarray*}
-\frac{\partial T^{^{-1}}}{\partial\xi} & = & -\Ad T\end{eqnarray*}
-
-\end_inset
-
-
-\begin_inset Formula \begin{eqnarray*}
-\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}} & = & \Ad{T_{2}^{^{-1}}}=1\mbox{ and }\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{3}\end{eqnarray*}
-
-\end_inset
-
-
-\begin_inset Formula \begin{eqnarray*}
-\frac{\partial\left(T_{1}^{-1}T_{2}\right)}{\partial\xi_{1}} & = & -\Ad{T_{2}^{^{-1}}T_{1}}=-\Ad{between(T_{2},T_{1})}\mbox{ and }\frac{\partial\left(T_{1}^{-1}T_{2}\right)}{\partial\xi_{2}}=I_{3}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-The derivatives of 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
-
-Hence, the final derivative of the group action is
-\begin_inset Formula \[
-\deriv{q(\xi)}{\xi}=R\left[\begin{array}{cc}
-I_{2} & R_{\pi/2}p\end{array}\right]=\left[\begin{array}{cc}
-R & R_{\pi/2}q\end{array}\right]\]
-
-\end_inset
-
-The derivative of the inverse action 
-\begin_inset Formula $\hat{q}=T^{-1}\hat{p}$
-\end_inset
-
- is given by 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:inverseAction"
-
-\end_inset
-
-, specialized to 
-\begin_inset Formula $\SEtwo$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\deriv{\left(T^{-1}\hat{p}\right)}{\xi}=-T^{-1}\deriv{\left(\Ad T\hat{\xi}\right)\hat{p}}{\xi}\]
-
-\end_inset
-
-where the velocity now is
-\begin_inset Formula \[
-\left(\Ad T\hat{\xi}\right)\hat{p}=\left[\begin{array}{cc}
-\skew{\omega} & Rv-\omega R_{\pi/2}t\\
-0 & 0\end{array}\right]\left[\begin{array}{c}
-p\\
-1\end{array}\right]=\left[\begin{array}{c}
-Rv+R_{\pi/2}(p-t)\omega\\
-0\end{array}\right]\]
-
-\end_inset
-
-and hence
-\begin_inset Formula \[
-\deriv{q(\xi)}{\xi}=-R^{T}\left[\begin{array}{cc}
-R & R_{\pi/2}(p-t)\end{array}\right]=\left[\begin{array}{cc}
--I_{2} & -R_{\pi/2}q\end{array}\right]\]
-
-\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
-Derivatives of Mappings
-\end_layout
-
-\begin_layout Standard
-Hence, we are now in a position to simply posit the derivative of 
-\series bold
-inverse
-\series default
-,
-\begin_inset Formula \begin{eqnarray*}
-\Skew{\omega'} & = & \Ad R\left(\Skew{-\omega}\right)=\Skew{R(-\omega)}\\
-\frac{\partial R^{T}}{\partial\omega} & = & -R\end{eqnarray*}
-
-\end_inset
-
-
-\series bold
-compose,
-\series default
-
-\begin_inset Formula \[
-\Skew{\omega'}=\Ad{R_{2}^{T}}\left(\Skew{\omega}\right)=\Skew{R_{2}^{T}\omega}\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-\frac{\partial\left(R_{1}R_{2}\right)}{\partial\omega_{1}}=R_{2}^{T}\mbox{ and }\frac{\partial\left(R_{1}R_{2}\right)}{\partial\omega_{2}}=I_{3}\]
-
-\end_inset
-
-
-\series bold
-between
-\series default
- in its first argument,
-\begin_inset Formula \begin{eqnarray*}
-\Skew{\omega'} & = & \Ad{R_{2}^{T}R_{1}}\left(\Skew{-\omega}\right)=\Skew{R_{2}^{T}R_{1}(-\omega)}\\
-\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\omega_{1}} & = & -R_{2}^{T}R_{1}=-between(R_{2},R_{1})\end{eqnarray*}
-
-\end_inset
-
-and between in its second argument,
-\begin_inset Formula \begin{eqnarray*}
-\frac{\partial\left(R_{1}^{T}R_{2}\right)}{\partial\omega_{2}} & = & I_{3}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Derivatives of 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 (following the exposition in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Derivatives-of-Actions"
-
-\end_inset
-
-):
-\begin_inset Formula \[
-\deriv{q(\omega)}{\omega}=R\deriv{}{\omega}\left(e^{\Skew{\omega}}p\right)=R\deriv{}{\omega}\left(\Skew{\omega}p\right)=RH_{p}\]
-
-\end_inset
-
-To calculate 
-\begin_inset Formula $H_{p}$
-\end_inset
-
- we make use of 
-\begin_inset Formula \[
-\Skew{\omega}p=\omega\times p=-p\times\omega=\Skew{-p}\omega\]
-
-\end_inset
-
-Hence, the final derivative of an action in its first argument is
-\begin_inset Formula \[
-\deriv{q(\omega)}{\omega}=RH_{p}=R\Skew{-p}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The derivative of the inverse action is given by 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:inverseAction"
-
-\end_inset
-
-, specialized to 
-\begin_inset Formula $\SOthree$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\deriv{q(\omega)}{\omega}=-R^{T}\deriv{\left(\Skew{R\omega\mbox{ }}p\right)}{\omega}=R^{T}\Skew pR=\Skew{R^{T}p}\]
-
-\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 Subsection
-Derivatives of Mappings
-\end_layout
-
-\begin_layout Standard
-Hence, as with 
-\begin_inset Formula $\SOthree$
-\end_inset
-
-, we are now in a position to simply posit the derivative of 
-\series bold
-inverse
-\series default
-,
-\begin_inset Formula \[
-\frac{\partial T^{-1}}{\partial\xi}=\Ad T=-\left[\begin{array}{cc}
-R & 0\\
-\Skew tR & R\end{array}\right]\]
-
-\end_inset
-
-(but unit test on the above fails !!!), 
-\series bold
-compose
-\series default
- in its first argument,
-\begin_inset Formula \[
-\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{2}^{-1}}=\left[\begin{array}{cc}
-R_{2}^{T} & 0\\
-\Skew{-R_{2}^{T}t_{2}}R_{2}^{T} & R_{2}^{T}\end{array}\right]=\left[\begin{array}{cc}
-R_{2}^{T} & 0\\
-R_{2}^{T}\Skew{-t_{2}} & R_{2}^{T}\end{array}\right]\]
-
-\end_inset
-
-compose in its second argument,
-\begin_inset Formula \[
-\frac{\partial\left(T_{1}T_{2}\right)}{\partial\xi_{2}}=I_{6}\]
-
-\end_inset
-
-
-\series bold
-between
-\series default
- in its first argument,
-\begin_inset Formula \[
-\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}}=\Ad{T_{21}}=-\left[\begin{array}{cc}
-R & 0\\
-\Skew tR & R\end{array}\right]\]
-
-\end_inset
-
-with 
-\begin_inset Formula \[
-T_{12}=\left[\begin{array}{cc}
-R & t\\
-0 & 1\end{array}\right]=T_{1}^{^{-1}}T_{2}=between(T_{2},T_{1})\]
-
-\end_inset
-
-and between in its second argument,
-\begin_inset Formula \begin{eqnarray*}
-\frac{\partial\left(T_{1}^{^{-1}}T_{2}\right)}{\partial\xi_{1}} & = & I_{6}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-The derivatives of 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 \[
-\hat{q}=\left[\begin{array}{c}
-q\\
-1\end{array}\right]=\left[\begin{array}{cc}
-R & t\\
-0 & 1\end{array}\right]\left[\begin{array}{c}
-p\\
-1\end{array}\right]=T\hat{p}\]
-
-\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 (following the exposition in Section 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "sec:Derivatives-of-Actions"
-
-\end_inset
-
-):
-\begin_inset Formula \[
-\deriv{\hat{q}(\xi)}{\xi}=T\deriv{}{\xi}\left(\xihat\hat{p}\right)=TH_{p}\]
-
-\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 \[
-\omega\times p+v=-p\times\omega+v=\left[\begin{array}{cc}
--\Skew p & I_{3}\end{array}\right]\left[\begin{array}{c}
-\omega\\
-v\end{array}\right]=H_{p}\xi\]
-
-\end_inset
-
-Hence, the final derivative of the group action is
-\begin_inset Formula \[
-\deriv{\hat{q}(\xi)}{\xi}=T\hat{H}_{p}=\left[\begin{array}{cc}
-R & t\\
-0 & 1\end{array}\right]\left[\begin{array}{cc}
-\Skew{-p} & I_{3}\\
-0 & 0\end{array}\right]\]
-
-\end_inset
-
-in homogenous coordinates.
- In 
-\begin_inset Formula $\Rthree$
-\end_inset
-
- this becomes:
-\begin_inset Formula \[
-\deriv{q(\xi)}{\xi}=R\left[\begin{array}{cc}
--\Skew p & I_{3}\end{array}\right]\]
-
-\end_inset
-
-The derivative of the inverse action 
-\begin_inset Formula $T^{-1}p$
-\end_inset
-
- is given by formula 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "eq:inverseAction"
-
-\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 \[
-\deriv{\hat{q}(\xi)}{\xi}=-H_{q}=\left[\begin{array}{cc}
-\Skew q & -I_{3}\end{array}\right]\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Pose3 (gtsam, old-style exmap)
-\end_layout
-
-\begin_layout Standard
-In the old-style, we have 
-\end_layout
-
-\begin_layout Standard
-
-\family roman
-\series medium
-\shape up
-\size normal
-\emph off
-\bar no
-\noun off
-\color none
-\begin_inset Formula $R'=R(I+\Omega)$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $t'=t+dt$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In this case, the derivative of 
-\series bold
-\emph on
-transform_from
-\series default
-\emph default
-, 
-\begin_inset Formula $Rx+t$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\frac{\partial(R(I+\Omega)x+t)}{\partial\omega}=\frac{\partial(R\Omega x)}{\partial\omega}=\frac{\partial(R\left(\omega\times x\right))}{\partial\omega}=R\Skew{-x}\]
-
-\end_inset
-
-and with respect to 
-\begin_inset Formula $dt$
-\end_inset
-
- is easy:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\frac{\partial(Rx+t+dt)}{\partial dt}=I\]
-
-\end_inset
-
-The derivative of 
-\series bold
-\emph on
-transform_to
-\series default
-\emph default
-, 
-\begin_inset Formula $inv(R)(x-t)$
-\end_inset
-
- we can obtain using the chain rule:
-\begin_inset Formula \[
-\frac{\partial(inv(R)(x-t))}{\partial\omega}=\frac{\partial unrot(R,(x-t))}{\partial\omega}=skew(R^{T}\left(x-t\right))\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-and with respect to 
-\begin_inset Formula $dt$
-\end_inset
-
- is easy:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\frac{\partial(R^{T}(x-t-dt))}{\partial dt}=-R^{T}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-2D Line Segments (Ocaml)
-\end_layout
-
-\begin_layout Standard
-The error between an infinite line 
-\begin_inset Formula $(a,b,c)$
-\end_inset
-
- and a 2D line segment 
-\begin_inset Formula $((x1,y1),(x2,y2))$
-\end_inset
-
- is defined in Line3.ml.
-\end_layout
-
-\begin_layout Section
-Line3vd (Ocaml)
-\end_layout
-
-\begin_layout Standard
-One representation of a line is through 2 vectors 
-\begin_inset Formula $(v,d)$
-\end_inset
-
-, where 
-\begin_inset Formula $v$
-\end_inset
-
- is the direction and the vector 
-\begin_inset Formula $d$
-\end_inset
-
- points from the orgin to the closest point on the line.
-\end_layout
-
-\begin_layout Standard
-In this representation, transforming a 3D line from a world coordinate frame
- to a camera at 
-\begin_inset Formula $(R_{w}^{c},t^{w})$
-\end_inset
-
- is done by
-\begin_inset Formula \[
-v^{c}=R_{w}^{c}v^{w}\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-d^{c}=R_{w}^{c}\left(d^{w}+(t^{w}v^{w})v^{w}-t^{w}\right)\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Line3 (Ocaml)
-\end_layout
-
-\begin_layout Standard
-For 3D lines, we use a parameterization due to C.J.
- Taylor, using a rotation matrix 
-\begin_inset Formula $R$
-\end_inset
-
- and 2 scalars 
-\begin_inset Formula $a$
-\end_inset
-
- and 
-\begin_inset Formula $b$
-\end_inset
-
-.
- The line direction 
-\begin_inset Formula $v$
-\end_inset
-
- is simply the Z-axis of the rotated frame, i.e., 
-\begin_inset Formula $v=R_{3}$
-\end_inset
-
-, while the vector 
-\begin_inset Formula $d$
-\end_inset
-
- is given by 
-\begin_inset Formula $d=aR_{1}+bR_{2}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Now, we will 
-\emph on
-not
-\emph default
- use the incremental rotation scheme we used for rotations: because the
- matrix R translates from the line coordinate frame to the world frame,
- we need to apply the incremental rotation on the right-side:
-\begin_inset Formula \[
-R'=R(I+\Omega)\]
-
-\end_inset
-
-Projecting a line to 2D can be done easily, as both 
-\begin_inset Formula $v$
-\end_inset
-
- and 
-\begin_inset Formula $d$
-\end_inset
-
- are also the 2D homogenous coordinates of two points on the projected line,
- and hence we have
-\begin_inset Formula \begin{eqnarray*}
-l & = & v\times d\\
- & = & R_{3}\times\left(aR_{1}+bR_{2}\right)\\
- & = & a\left(R_{3}\times R_{1}\right)+b\left(R_{3}\times R_{2}\right)\\
- & = & aR_{2}-bR_{1}\end{eqnarray*}
-
-\end_inset
-
-This can be written as a rotation of a point,
-\begin_inset Formula \[
-l=R\left(\begin{array}{c}
--b\\
-a\\
-0\end{array}\right)\]
-
-\end_inset
-
-but because the incremental rotation is now done on the right, we need to
- figure out the derivatives again:
-\begin_inset Formula \begin{equation}
-\frac{\partial(R(I+\Omega)x)}{\partial\omega}=\frac{\partial(R\Omega x)}{\partial\omega}=R\frac{\partial(\Omega x)}{\partial\omega}=R\Skew{-x}\label{eq:rotateRight}\end{equation}
-
-\end_inset
-
-and hence the derivative of the projection 
-\begin_inset Formula $l$
-\end_inset
-
- with respect to the rotation matrix 
-\begin_inset Formula $R$
-\end_inset
-
-of the 3D line is 
-\begin_inset Formula \begin{equation}
-\frac{\partial(l)}{\partial\omega}=R\Skew{\left(\begin{array}{c}
-b\\
--a\\
-0\end{array}\right)}=\left[\begin{array}{ccc}
-aR_{3} & bR_{3} & -(aR_{1}+bR_{2})\end{array}\right]\end{equation}
-
-\end_inset
-
-or the 
-\begin_inset Formula $a,b$
-\end_inset
-
- scalars:
-\begin_inset Formula \[
-\frac{\partial(l)}{\partial a}=R_{2}\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-\frac{\partial(l)}{\partial b}=-R_{1}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Transforming a 3D line 
-\begin_inset Formula $(R,(a,b))$
-\end_inset
-
- from a world coordinate frame to a camera frame 
-\begin_inset Formula $(R_{w}^{c},t^{w})$
-\end_inset
-
- is done by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-R'=R_{w}^{c}R\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-a'=a-R_{1}^{T}t^{w}\]
-
-\end_inset
-
-
-\begin_inset Formula \[
-b'=b-R_{2}^{T}t^{w}\]
-
-\end_inset
-
-Again, we need to redo the derivatives, as R is incremented from the right.
- The first argument is incremented from the left, but the result is incremented
- on the right:
-\begin_inset Formula \begin{eqnarray*}
-R'(I+\Omega')=(AB)(I+\Omega') & = & (I+\Skew{S\omega})AB\\
-I+\Omega' & = & (AB)^{T}(I+\Skew{S\omega})(AB)\\
-\Omega' & = & R'^{T}\Skew{S\omega}R'\\
-\Omega' & = & \Skew{R'^{T}S\omega}\\
-\omega' & = & R'^{T}S\omega\end{eqnarray*}
-
-\end_inset
-
-For the second argument 
-\begin_inset Formula $R$
-\end_inset
-
- we now simply have:
-\begin_inset Formula \begin{eqnarray*}
-AB(I+\Omega') & = & AB(I+\Omega)\\
-\Omega' & = & \Omega\\
-\omega' & = & \omega\end{eqnarray*}
-
-\end_inset
-
-The scalar derivatives can be found by realizing that 
-\begin_inset Formula \[
-\left(\begin{array}{c}
-a'\\
-b'\\
-...\end{array}\right)=\left(\begin{array}{c}
-a\\
-b\\
-0\end{array}\right)-R^{T}t^{w}\]
-
-\end_inset
-
-where we don't care about the third row.
- Hence
-\begin_inset Formula \[
-\frac{\partial(\left(R(I+\Omega_{2})\right)^{T}t^{w})}{\partial\omega}=-\frac{\partial(\Omega_{2}R^{T}t^{w})}{\partial\omega}=-\Skew{R^{T}t^{w}}=\left[\begin{array}{ccc}
-0 & R_{3}^{T}t^{w} & -R_{2}^{T}t^{w}\\
--R_{3}^{T}t^{w} & 0 & R_{1}^{T}t^{w}\\
-... & ... & 0\end{array}\right]\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-
-\series bold
-Aligning 3D Scans
-\end_layout
-
-\begin_layout Standard
-Below is the explanaition underlying Pose3.align, i.e.
- aligning two point clouds using SVD.
- Inspired but modified from CVOnline...
-\end_layout
-
-\begin_layout Standard
-
-\emph on
-Our
-\emph default
- model is
-\begin_inset Formula \[
-p^{c}=R\left(p^{w}-t\right)\]
-
-\end_inset
-
-i.e., 
-\begin_inset Formula $R$
-\end_inset
-
- is from camera to world, and 
-\begin_inset Formula $t$
-\end_inset
-
- is the camera location in world coordinates.
- The objective function is
-\begin_inset Formula \begin{equation}
-\frac{1}{2}\sum\left(p^{c}-R(p^{w}-t)\right)^{2}=\frac{1}{2}\sum\left(p^{c}-Rp^{w}+Rt\right)^{2}=\frac{1}{2}\sum\left(p^{c}-Rp^{w}-t'\right)^{2}\label{eq:J}\end{equation}
-
-\end_inset
-
-where 
-\begin_inset Formula $t'=-Rt$
-\end_inset
-
- is the location of the origin in the camera frame.
- Taking the derivative with respect to 
-\begin_inset Formula $t'$
-\end_inset
-
- and setting to zero we have
-\begin_inset Formula \[
-\sum\left(p^{c}-Rp^{w}-t'\right)=0\]
-
-\end_inset
-
-or
-\begin_inset Formula \begin{equation}
-t'=\frac{1}{n}\sum\left(p^{c}-Rp^{w}\right)=\bar{p}^{c}-R\bar{p}^{w}\label{eq:t}\end{equation}
-
-\end_inset
-
-here 
-\begin_inset Formula $\bar{p}^{c}$
-\end_inset
-
- and 
-\begin_inset Formula $\bar{p}^{w}$
-\end_inset
-
- are the point cloud centroids.
- Substituting back into 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:J"
-
-\end_inset
-
-, we get
-\begin_inset Formula \[
-\frac{1}{2}\sum\left(p^{c}-R(p^{w}-t)\right)^{2}=\frac{1}{2}\sum\left(\left(p^{c}-\bar{p}^{c}\right)-R\left(p^{w}-\bar{p}^{w}\right)\right)^{2}=\frac{1}{2}\sum\left(\hat{p}^{c}-R\hat{p}^{w}\right)^{2}\]
-
-\end_inset
-
-Now, to minimize the above it suffices to maximize (see CVOnline) 
-\begin_inset Formula \[
-\mathop{trace}\left(R^{T}C\right)\]
-
-\end_inset
-
-where 
-\begin_inset Formula $C=\sum\hat{p}^{c}\left(\hat{p}^{w}\right)^{T}$
-\end_inset
-
- is the correlation matrix.
- Intuitively, the cloud of points is rotated to align with the principal
- axes.
- This can be achieved by SVD decomposition on 
-\begin_inset Formula $C$
-\end_inset
-
-
-\begin_inset Formula \[
-C=USV^{T}\]
-
-\end_inset
-
-and setting 
-\begin_inset Formula \[
-R=UV^{T}\]
-
-\end_inset
-
-Clearly, from 
-\begin_inset CommandInset ref
-LatexCommand eqref
-reference "eq:t"
-
-\end_inset
-
- we then also recover the optimal 
-\begin_inset Formula $t$
-\end_inset
-
- as 
-\begin_inset Formula \[
-t=\bar{p}^{w}-R^{T}\bar{p}^{c}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset CommandInset bibtex
-LatexCommand bibtex
-bibfiles "/Users/dellaert/papers/refs"
-options "plain"
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
diff --git a/doc/math.pdf b/doc/math.pdf
deleted file mode 100644
index 98e3fd758..000000000
Binary files a/doc/math.pdf and /dev/null differ
diff --git a/gtsam-broken.h b/gtsam-broken.h
deleted file mode 100644
index dd4ec379c..000000000
--- a/gtsam-broken.h
+++ /dev/null
@@ -1,144 +0,0 @@
-// These are currently broken
-// Solve by parsing a namespace pose2SLAM::Values and making a Pose2SLAMValues class
-// We also have to solve the shared pointer mess to avoid duplicate methods
-
-class GaussianFactor {
-	GaussianFactor(string key1,
-			Matrix A1,
-			Vector b_in,
-			const SharedDiagonal& model);
-	GaussianFactor(string key1,
-			Matrix A1,
-			string key2,
-			Matrix A2,
-			Vector b_in,
-			const SharedDiagonal& model);
-	GaussianFactor(string key1,
-			Matrix A1,
-			string key2,
-			Matrix A2,
-			string key3,
-			Matrix A3,
-			Vector b_in,
-			const SharedDiagonal& model);
-	bool involves(string key) const;
-	Matrix getA(string key) const;
-	pair<Matrix,Vector> matrix(const Ordering& ordering) const;
-	pair<GaussianConditional*,GaussianFactor*> eliminate(string key) const;
-};
-
-class GaussianConditional {
-	GaussianConditional(string key,
-			Vector d,
-			Matrix R,
-			Vector sigmas);
-	GaussianConditional(string key,
-			Vector d,
-			Matrix R,
-			string name1,
-			Matrix S,
-			Vector sigmas);
-	GaussianConditional(string key,
-			Vector d,
-			Matrix R,
-			string name1,
-			Matrix S,
-			string name2,
-			Matrix T,
-			Vector sigmas);
-	void add(string key, Matrix S);
-};
-
-class GaussianFactorGraph {
-	GaussianConditional* eliminateOne(string key);
-	GaussianBayesNet* eliminate_(const Ordering& ordering);
-	VectorValues* optimize_(const Ordering& ordering);
-	pair<Matrix,Vector> matrix(const Ordering& ordering) const;
-	Matrix sparse(const Ordering& ordering) const;
-	VectorValues* steepestDescent_(const VectorValues& x0) const;
-	VectorValues* conjugateGradientDescent_(const VectorValues& x0) const;
-};
-
-
-class Pose2Values{
-	Pose2Values();
-	Pose2 get(string key) const;
-	void insert(string name, const Pose2& val);
-	void print(string s) const;
-	void clear();
-	int size();
-};
-
-class Pose2Factor {
-	Pose2Factor(string key1, string key2,
-			const Pose2& measured, Matrix measurement_covariance);
-	void print(string name) const;
-	double error(const Pose2Values& c) const;
-	size_t size() const;
-	GaussianFactor* linearize(const Pose2Values& config) const;
-};
-
-class Pose2Graph{
-	Pose2Graph();
-	void print(string s) const;
-	GaussianFactorGraph* linearize_(const Pose2Values& config) const;
-	void push_back(Pose2Factor* factor);
-};
-
-class Ordering{
-	Ordering(string key);
-	Ordering subtract(const Ordering& keys) const;
-	void unique ();
-	void reverse ();
-};
-
-
-class SymbolicFactor{
-	SymbolicFactor(const Ordering& keys);
-	void print(string s) const;
-};
-
-
-class VectorValues {
-	void insert(string name, Vector val);
-	Vector get(string name) const;
-	bool contains(string name) const;
-};
-
-class Simulated2DPosePrior {
-	GaussianFactor* linearize(const Simulated2DValues& config) const;
-};
-
-class Simulated2DOrientedPosePrior {
-	GaussianFactor* linearize(const Simulated2DOrientedValues& config) const;
-};
-
-class Simulated2DPointPrior {
-	GaussianFactor* linearize(const Simulated2DValues& config) const;
-};
-
-class Simulated2DOdometry {
-	GaussianFactor* linearize(const Simulated2DValues& config) const;
-};
-
-class Simulated2DOrientedOdometry {
-	GaussianFactor* linearize(const Simulated2DOrientedValues& config) const;
-};
-
-class Simulated2DMeasurement {
-	GaussianFactor* linearize(const Simulated2DValues& config) const;
-};
-
-class Pose2SLAMOptimizer {
-	Pose2SLAMOptimizer(string dataset_name);
-	void print(string s) const;
-	void update(Vector x) const;
-	Vector optimize() const;
-	double error() const;
-	Matrix a1() const;
-	Matrix a2() const;
-	Vector b1() const;
-	Vector b2() const;
-};
-
-
diff --git a/gtsam.h b/gtsam.h
deleted file mode 100644
index 268b8b2fe..000000000
--- a/gtsam.h
+++ /dev/null
@@ -1,160 +0,0 @@
-class SharedGaussian {
-	SharedGaussian(Matrix covariance);
-	SharedGaussian(Vector sigmas);
-};
-
-class SharedDiagonal {
-	SharedDiagonal(Vector sigmas);
-};
-
-class Ordering {
-  Ordering();
-  void print(string s) const;
-  bool equals(const Ordering& ord, double tol) const;
-  void push_back(string s);
-};
-
-
-class VectorValues {
-  VectorValues();
-  VectorValues(size_t nVars, size_t varDim);
-  void print(string s) const;
-  bool equals(const VectorValues& expected, double tol) const;
-  size_t size() const;
-};
-
-class GaussianFactor {
-	void print(string s) const;
-	bool equals(const GaussianFactor& lf, double tol) const;
-	bool empty() const;
-	Vector getb() const;
-	double error(const VectorValues& c) const;
-};
-
-class GaussianFactorSet {
-  GaussianFactorSet();
-  void push_back(GaussianFactor* factor);
-};
-
-class GaussianConditional {
-  GaussianConditional();
-  void print(string s) const;
-  bool equals(const GaussianConditional &cg, double tol) const;
-  Vector solve(const VectorValues& x);
-};
-
-class GaussianBayesNet {
-  GaussianBayesNet();
-  void print(string s) const;
-  bool equals(const GaussianBayesNet& cbn, double tol) const;
-  void push_back(GaussianConditional* conditional);
-  void push_front(GaussianConditional* conditional);
-};
-
-class GaussianFactorGraph {
-  GaussianFactorGraph();
-  void print(string s) const;
-  bool equals(const GaussianFactorGraph& lfgraph, double tol) const;
-
-  size_t size() const;
-  void push_back(GaussianFactor* ptr_f);
-  double error(const VectorValues& c) const;
-  double probPrime(const VectorValues& c) const;
-  void combine(const GaussianFactorGraph& lfg);
-};
-
-class Point2 {
-  Point2();
-  Point2(double x, double y);
-  void print(string s) const;
-  double x();
-  double y();
-};
-
-class Point3 {
-  Point3();
-  Point3(double x, double y, double z);
-  Point3(Vector v);
-  void print(string s) const;
-  Vector vector() const;
-  double x();
-  double y();
-  double z();
-}; 
-
-class Pose2 {
-  Pose2();
-  Pose2(const Pose2& pose);
-  Pose2(double x, double y, double theta);
-  Pose2(double theta, const Point2& t);
-  Pose2(const Rot2& r, const Point2& t);
-  void print(string s) const;
-  bool equals(const Pose2& pose, double tol) const;
-  double x() const;
-  double y() const;
-  double theta() const;
-  size_t dim() const;
-  Pose2 expmap(const Vector& v) const;
-  Vector logmap(const Pose2& pose) const;
-  Point2 t() const;
-  Rot2 r() const;
-};
-
-class Simulated2DValues {
-	Simulated2DValues();
-  void print(string s) const;
-	void insertPose(int i, const Point2& p);
-	void insertPoint(int j, const Point2& p);
-	int nrPoses() const;
-	int nrPoints() const;
-	Point2* pose(int i);
-	Point2* point(int j);
-};
-
-class Simulated2DOrientedValues {
-	Simulated2DOrientedValues();
-  void print(string s) const;
-	void insertPose(int i, const Pose2& p);
-	void insertPoint(int j, const Point2& p);
-	int nrPoses() const;
-	int nrPoints() const;
-	Pose2* pose(int i);
-	Point2* point(int j);
-};
-
-class Simulated2DPosePrior {
-	Simulated2DPosePrior(Point2& mu, const SharedDiagonal& model, int i);
-  void print(string s) const;
-  double error(const Simulated2DValues& c) const;
-};
-
-class Simulated2DOrientedPosePrior {
-	Simulated2DOrientedPosePrior(Pose2& mu, const SharedDiagonal& model, int i);
-  void print(string s) const;
-  double error(const Simulated2DOrientedValues& c) const;
-};
-
-class Simulated2DPointPrior {
-	Simulated2DPointPrior(Point2& mu, const SharedDiagonal& model, int i);
-  void print(string s) const;
-  double error(const Simulated2DValues& c) const;
-};
-
-class Simulated2DOdometry {
-  Simulated2DOdometry(Point2& mu, const SharedDiagonal& model, int i1, int i2);
-  void print(string s) const;
-  double error(const Simulated2DValues& c) const;
-};
-
-class Simulated2DOrientedOdometry {
-	Simulated2DOrientedOdometry(Pose2& mu, const SharedDiagonal& model, int i1, int i2);
-  void print(string s) const;
-  double error(const Simulated2DOrientedValues& c) const;
-};
-
-class Simulated2DMeasurement {
-  Simulated2DMeasurement(Point2& mu, const SharedDiagonal& model, int i, int j);
-  void print(string s) const;
-  double error(const Simulated2DValues& c) const;
-};
-
diff --git a/wrap/.cvsignore b/wrap/.cvsignore
deleted file mode 100644
index e9c87a8fc..000000000
--- a/wrap/.cvsignore
+++ /dev/null
@@ -1,5 +0,0 @@
-Makefile
-Makefile.in
-wrap
-.deps
-.DS_Store
diff --git a/wrap/Argument.cpp b/wrap/Argument.cpp
deleted file mode 100644
index e44b3033c..000000000
--- a/wrap/Argument.cpp
+++ /dev/null
@@ -1,85 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Argument.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-#include <sstream>
-#include <boost/foreach.hpp>
-
-#include "Argument.h"
-
-using namespace std;
-
-/* ************************************************************************* */
-void Argument::matlab_unwrap(ofstream& ofs, 
-			     const string& matlabName) 
-{ 
-  // the templated unwrap function returns a pointer
-  // example: double tol = unwrap< double >(in[2]);
-  ofs << "  ";
-
-  if (is_ptr)      
-    ofs << "shared_ptr<" << type << "> " << name << " = unwrap_shared_ptr< ";
-  else if (is_ref) 
-    ofs <<                         type << "& " << name << " = *unwrap_shared_ptr< ";
-  else
-    ofs <<                         type << " "  << name << " = unwrap< ";
-
-  ofs << type << " >(" << matlabName;
-  if (is_ptr || is_ref) ofs << ", \"" << type << "\"";
-  ofs << ");" << endl;
-}
-
-/* ************************************************************************* */
-string ArgumentList::types() { 
-  string str;
-  bool first=true;
-  BOOST_FOREACH(Argument arg, *this)  {
-    if (!first) str += ","; str += arg.type; first=false;
-  }
-  return str;
-}
-
-/* ************************************************************************* */
-string ArgumentList::signature() { 
-  string str;
-  BOOST_FOREACH(Argument arg, *this)
-    str += arg.type[0];
-  return str;
-}
-
-/* ************************************************************************* */
-string ArgumentList::names() { 
-  string str;
-  bool first=true;
-  BOOST_FOREACH(Argument arg, *this) {
-    if (!first) str += ","; str += arg.name; first=false;
-  }
-  return str;
-}
-
-/* ************************************************************************* */
-void ArgumentList::matlab_unwrap(ofstream& ofs, int start) { 
-  int index = start;
-  BOOST_FOREACH(Argument arg, *this) {
-    stringstream buf;
-    buf << "in[" << index << "]";
-    arg.matlab_unwrap(ofs,buf.str());
-    index++;
-  }
-}
-
-/* ************************************************************************* */
diff --git a/wrap/Argument.h b/wrap/Argument.h
deleted file mode 100644
index 73c3ca12f..000000000
--- a/wrap/Argument.h
+++ /dev/null
@@ -1,51 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Argument.h
- * brief: arguments to constructors and methods
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <string>
-#include <list>
-
-// Argument class
-struct Argument {
-  bool is_const, is_ref, is_ptr;
-  std::string type;
-  std::string name;
-Argument() : is_const(false), is_ref(false), is_ptr(false) {}
-
-  // MATLAB code generation:
-  void matlab_unwrap(std::ofstream& ofs, 
-		     const std::string& matlabName); // MATLAB to C++
-};
-
-// Argument list
-struct ArgumentList : public std::list<Argument> {
-  std::list<Argument> args;
-  std::string types    ();
-  std::string signature();
-  std::string names    ();
-
-  // MATLAB code generation:
-
-  /**
-   * emit code to unwrap arguments
-   * @param ofs output stream
-   * @param start initial index for input array, set to 1 for method
-   */
-  void matlab_unwrap(std::ofstream& ofs, int start=0); // MATLAB to C++
-};
-
diff --git a/wrap/Class.cpp b/wrap/Class.cpp
deleted file mode 100644
index de1620733..000000000
--- a/wrap/Class.cpp
+++ /dev/null
@@ -1,85 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Class.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-
-#include <boost/foreach.hpp>
-
-#include "Class.h"
-#include "utilities.h"
-
-using namespace std;
-
-/* ************************************************************************* */
-void Class::matlab_proxy(const string& classFile) {
-  // open destination classFile
-  ofstream ofs(classFile.c_str());
-  if(!ofs) throw CantOpenFile(classFile);
-  if(verbose_) cerr << "generating " << classFile << endl;
-
-  // emit class proxy code
-  emit_header_comment(ofs,"%");
-  ofs << "classdef " << name << endl;
-  ofs << "  properties" << endl;
-  ofs << "    self = 0" << endl;
-  ofs << "  end" << endl;
-  ofs << "  methods" << endl;
-  ofs << "    function obj = " << name << "(varargin)" << endl;
-  BOOST_FOREACH(Constructor c, constructors)
-    c.matlab_proxy_fragment(ofs,name);
-  ofs << "      if nargin ~= 13 && obj.self == 0, error('" << name << " constructor failed'); end" << endl;
-  ofs << "    end" << endl;
-  ofs << "    function display(obj), obj.print(''); end" << endl;
-  ofs << "    function disp(obj), obj.display; end" << endl;
-  ofs << "  end" << endl;
-  ofs << "end" << endl;
-
-  // close file
-  ofs.close();
-}
-
-/* ************************************************************************* */
-void Class::matlab_constructors(const string& toolboxPath,const string& nameSpace) {
-  BOOST_FOREACH(Constructor c, constructors) {
-    c.matlab_mfile  (toolboxPath, name);
-    c.matlab_wrapper(toolboxPath, name, nameSpace);
-  }
-}
-
-/* ************************************************************************* */
-void Class::matlab_methods(const string& classPath, const string& nameSpace) {
-  BOOST_FOREACH(Method m, methods) {
-    m.matlab_mfile  (classPath);
-    m.matlab_wrapper(classPath, name, nameSpace);
-  }
-}
-
-/* ************************************************************************* */
-void Class::matlab_make_fragment(ofstream& ofs, 
-				 const string& toolboxPath,
-				 const string& mexFlags) 
-{
-  string mex = "mex " + mexFlags + " ";
-  BOOST_FOREACH(Constructor c, constructors)
-    ofs << mex << c.matlab_wrapper_name(name) << ".cpp" << endl;
-  ofs << endl << "cd @" << name << endl;
-  BOOST_FOREACH(Method m, methods)
-    ofs << mex << m.name << ".cpp" << endl;
-  ofs << endl;
-}
-
-/* ************************************************************************* */
diff --git a/wrap/Class.h b/wrap/Class.h
deleted file mode 100644
index f00812b95..000000000
--- a/wrap/Class.h
+++ /dev/null
@@ -1,45 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Class.h
- * brief: describe the C++ class that is being wrapped
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <string>
-#include <list>
-
-#include "Constructor.h"
-#include "Method.h"
-
-// Class has name, constructors, methods
-struct Class {
-  std::string name;
-  std::list<Constructor> constructors;
-  std::list<Method> methods;
-  bool verbose_;
-
-  Class(bool verbose=true) : verbose_(verbose) {}
-
-  // MATLAB code generation:
-  void matlab_proxy(const std::string& classFile);          // proxy class
-  void matlab_constructors(const std::string& toolboxPath, 
-			   const std::string& nameSpace);   // constructor wrappers
-  void matlab_methods(const std::string& classPath, 
-			   const std::string& nameSpace);   // method wrappers
-  void matlab_make_fragment(std::ofstream& ofs, 
-			    const std::string& toolboxPath, 
-			    const std::string& mexFlags);   // make fragment
-};
-
diff --git a/wrap/Constructor.cpp b/wrap/Constructor.cpp
deleted file mode 100644
index d26e16a07..000000000
--- a/wrap/Constructor.cpp
+++ /dev/null
@@ -1,100 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Constructor.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-
-#include <boost/foreach.hpp>
-
-#include "utilities.h"
-#include "Constructor.h"
-
-using namespace std;
-
-/* ************************************************************************* */
-string Constructor::matlab_wrapper_name(const string& className) {
-  string str = "new_" + className + "_" + args.signature();
-  return str;
-}
-
-/* ************************************************************************* */
-void Constructor::matlab_proxy_fragment(ofstream& ofs, const string& className) {
-  ofs << "      if nargin == " << args.size() << ", obj.self = " 
-      << matlab_wrapper_name(className) << "(";
-  bool first = true;
-  for(size_t i=0;i<args.size();i++) {
-    if (!first) ofs << ",";
-    ofs << "varargin{" << i+1 << "}";
-    first=false;
-  }
-  ofs << "); end" << endl;
-}
-
-/* ************************************************************************* */
-void Constructor::matlab_mfile(const string& toolboxPath, const string& className) {
-
-  string name = matlab_wrapper_name(className);
-
-  // open destination m-file
-  string wrapperFile = toolboxPath + "/" + name + ".m";
-  ofstream ofs(wrapperFile.c_str());
-  if(!ofs) throw CantOpenFile(wrapperFile);
-  if(verbose_) cerr << "generating " << wrapperFile << endl;
-
-  // generate code
-  emit_header_comment(ofs, "%");
-  ofs << "function result = " << name << "(obj";
-  if (args.size()) ofs << "," << args.names();
-  ofs << ")" << endl;
-  ofs << "  error('need to compile " << name << ".cpp');" << endl;
-  ofs << "end" << endl;
-
-  // close file
-  ofs.close();
-}
-
-/* ************************************************************************* */
-void Constructor::matlab_wrapper(const string& toolboxPath, 
-				 const string& className,
-				 const string& nameSpace) 
-{
-
-  string name = matlab_wrapper_name(className);
-
-  // open destination wrapperFile
-  string wrapperFile = toolboxPath + "/" + name + ".cpp";
-  ofstream ofs(wrapperFile.c_str());
-  if(!ofs) throw CantOpenFile(wrapperFile);
-  if(verbose_) cerr << "generating " << wrapperFile << endl;
-
-  // generate code
-  emit_header_comment(ofs, "//");
-  ofs << "#include <wrap/matlab.h>" << endl;
-  ofs << "#include <" << className << ".h>" << endl;
-  if (!nameSpace.empty()) ofs << "using namespace " << nameSpace << ";" << endl;
-  ofs << "void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])" << endl;
-  ofs << "{" << endl;
-  ofs << "  checkArguments(\"" << name << "\",nargout,nargin," << args.size() << ");" << endl;
-  args.matlab_unwrap(ofs); // unwrap arguments
-  ofs << "  " << className << "* self = new " << className << "(" << args.names() << ");" << endl;
-  ofs << "  out[0] = wrap_constructed(self,\"" << className << "\");" << endl;
-  ofs << "}" << endl;
-
-  // close file
-  ofs.close();
-}
-
-/* ************************************************************************* */
diff --git a/wrap/Constructor.h b/wrap/Constructor.h
deleted file mode 100644
index 0a15c06d1..000000000
--- a/wrap/Constructor.h
+++ /dev/null
@@ -1,43 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Constructor.h
- * brief: class describing a constructor + code generation
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <string>
-#include <list>
-
-#include "Argument.h"
-
-// Constructor class
-struct Constructor {
-  ArgumentList args;
-  bool verbose_;
-
-  Constructor(bool verbose=true) : verbose_(verbose) {}
-
-  // MATLAB code generation
-  // toolboxPath is main toolbox directory, e.g., ../matlab
-  // classFile is class proxy file, e.g., ../matlab/@Point2/Point2.m
-
-  std::string matlab_wrapper_name(const std::string& className);                     // wrapper name
-  void matlab_proxy_fragment(std::ofstream& ofs, const std::string& className);      // proxy class fragment
-  void matlab_mfile  (const std::string& toolboxPath, const std::string& className); // m-file
-  void matlab_wrapper(const std::string& toolboxPath, 
-		      const std::string& className, 
-		      const std::string& nameSpace); // wrapper
-};
-
diff --git a/wrap/Makefile.am b/wrap/Makefile.am
deleted file mode 100644
index 620dab26e..000000000
--- a/wrap/Makefile.am
+++ /dev/null
@@ -1,41 +0,0 @@
-common = utilities.cpp Argument.cpp Constructor.cpp Method.cpp Class.cpp Module.cpp
-
-check_PROGRAMS = testSpirit testWrap 
-testSpirit_SOURCES  = testSpirit.cpp
-testWrap_SOURCES = testWrap.cpp ${common}
-
-# generate local toolbox dir
-interfacePath = $(top_srcdir)
-moduleName = gtsam
-toolboxpath = ../toolbox
-nameSpace = "gtsam"
-mexFlags = "${BOOST_CPPFLAGS} -I${prefix}/include -I${prefix}/include/gtsam/linear -I${prefix}/include/gtsam/nonlinear -I${prefix}/include/gtsam/base -I${prefix}/include/gtsam/geometry -I${prefix}/include/gtsam/slam -L${exec_prefix}/lib -lgtsam"
-all:
-	./wrap ${interfacePath} ${moduleName} ${toolboxpath} ${nameSpace} ${mexFlags}
-
-# install the header files
-noinst_HEADERS = geometry.h utilities.h Argument.h Constructor.h Method.h Class.h Module.h wrap-matlab.h
-
-noinst_PROGRAMS = wrap
-wrap_SOURCES = ${common} wrap.cpp
-AM_CPPFLAGS = $(BOOST_CPPFLAGS) -I$(top_srcdir) -DTOPSRCDIR="\"$(top_srcdir)\""
-AM_CXXFLAGS = -MMD
-AM_LDFLAGS = $(BOOST_LDFLAGS) -L../CppUnitLite -lCppUnitLite
-
-TESTS = $(check_PROGRAMS)
-
-# install the headers and matlab toolbox
-install-exec-hook: all
-	install -d ${toolbox}/gtsam && \
-	cp -rf ../toolbox/* ${toolbox}/gtsam && \
-	install -d ${includedir}/wrap && \
-	cp -f ${srcdir}/wrap-matlab.h  ${includedir}/wrap/matlab.h
-
-# clean local toolbox dir
-clean:
-	@test -z "wrap" || rm -f wrap
-	@test -z "../toolbox" || rm -rf ../toolbox
-
-# rule to run an executable
-%.run: % libgtsam.la
-	./$^
diff --git a/wrap/Method.cpp b/wrap/Method.cpp
deleted file mode 100644
index cec6391bb..000000000
--- a/wrap/Method.cpp
+++ /dev/null
@@ -1,139 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Method.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-
-#include <boost/foreach.hpp>
-
-#include "Method.h"
-#include "utilities.h"
-
-using namespace std;
-
-/* ************************************************************************* */
-// auxiliary function to wrap an argument into a shared_ptr template
-/* ************************************************************************* */
-string maybe_shared_ptr(bool add, const string& type) {
-  string str = add? "shared_ptr<" : "";
-  str += type;
-  if (add) str += ">";
-  return str;
-}
-
-/* ************************************************************************* */
-string Method::return_type(bool add_ptr, pairing p) {
-  if (p==pair && returns_pair) {
-    string str = "pair< " + 
-      maybe_shared_ptr(add_ptr && returns_ptr, returns ) + ", " + 
-      maybe_shared_ptr(add_ptr && returns_ptr, returns2) + " >";
-    return str;
-  } else
-    return maybe_shared_ptr(add_ptr && returns_ptr, (p==arg2)? returns2 : returns);
-}
-
-/* ************************************************************************* */
-void Method::matlab_mfile(const string& classPath) {
-
-  // open destination m-file
-  string wrapperFile = classPath + "/" + name + ".m";
-  ofstream ofs(wrapperFile.c_str());
-  if(!ofs) throw CantOpenFile(wrapperFile);
-  if(verbose_) cerr << "generating " << wrapperFile << endl;
-
-  // generate code
-  emit_header_comment(ofs, "%");
-  ofs << "% usage: obj." << name << "(" << args.names() << ")" << endl;
-  string returnType = returns_pair? "[first,second]" : "result";
-  ofs << "function " << returnType << " = " << name << "(obj";
-  if (args.size()) ofs << "," << args.names();
-  ofs << ")" << endl;
-  ofs << "  error('need to compile " << name << ".cpp');" << endl;
-  ofs << "end" << endl;
-
-  // close file
-  ofs.close();
-}
-
-/* ************************************************************************* */
-void Method::matlab_wrapper(const string& classPath, 
-			    const string& className,
-			    const string& nameSpace) 
-{
-  // open destination wrapperFile
-  string wrapperFile = classPath + "/" + name + ".cpp";
-  ofstream ofs(wrapperFile.c_str());
-  if(!ofs) throw CantOpenFile(wrapperFile);
-  if(verbose_) cerr << "generating " << wrapperFile << endl;
-
-  // generate code
-
-  // header
-  emit_header_comment(ofs, "//");
-  ofs << "#include <wrap/matlab.h>\n";
-  ofs << "#include <" << className << ".h>\n";
-  if (!nameSpace.empty()) ofs << "using namespace " << nameSpace << ";" << endl;
-
-  // call
-  ofs << "void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])\n";
-  // start
-  ofs << "{\n";
-
-  // check arguments
-  // extra argument obj -> nargin-1 is passed !
-  // example: checkArguments("equals",nargout,nargin-1,2);
-  ofs << "  checkArguments(\"" << name << "\",nargout,nargin-1," << args.size() << ");\n";
-
-  // get class pointer
-  // example: shared_ptr<Test> = unwrap_shared_ptr< Test >(in[0], "Test");
-  ofs << "  shared_ptr<" << className << "> self = unwrap_shared_ptr< " << className 
-      << " >(in[0],\"" << className << "\");" << endl;
-
-  // unwrap arguments, see Argument.cpp
-  args.matlab_unwrap(ofs,1);
-
-  // call method
-  // example: bool result = self->return_field(t);
-  ofs << "  ";
-  if (returns!="void") 
-    ofs << return_type(true,pair) << " result = ";
-  ofs << "self->" << name << "(" << args.names() << ");\n";
-
-  // wrap result
-  // example: out[0]=wrap<bool>(result);
-  if (returns_pair) {
-    if (returns_ptr)
-      ofs << "  out[0] = wrap_shared_ptr(result.first,\"" << returns << "\");\n";
-    else
-      ofs << "  out[0] = wrap< " << return_type(true,arg1) << " >(result.first);\n";
-    if (returns_ptr2)
-      ofs << "  out[1] = wrap_shared_ptr(result.second,\"" << returns2 << "\");\n";
-    else
-      ofs << "  out[1] = wrap< " << return_type(true,arg2) << " >(result.second);\n";
-  } 
-  else if (returns_ptr)
-    ofs << "  out[0] = wrap_shared_ptr(result,\"" << returns << "\");\n";
-  else if (returns!="void")
-    ofs << "  out[0] = wrap< " << return_type(true,arg1) << " >(result);\n";
-
-  // finish
-  ofs << "}\n";
-
-  // close file
-  ofs.close();
-}
-
-/* ************************************************************************* */
diff --git a/wrap/Method.h b/wrap/Method.h
deleted file mode 100644
index 8666f74ac..000000000
--- a/wrap/Method.h
+++ /dev/null
@@ -1,46 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Method.h
- * brief: describes and generates code for methods
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <string>
-#include <list>
-
-#include "Argument.h"
-
-// Method class
-struct Method {
-  bool is_const;
-  ArgumentList args;
-  std::string returns, returns2, name;
-  bool returns_ptr, returns_ptr2, returns_pair;
-  bool verbose_;
-
-  Method(bool verbose=true) : returns_ptr(false), returns_ptr2(false), returns_pair(false), verbose_(verbose) {}
-
-  enum pairing {arg1, arg2, pair};
-  std::string return_type(bool add_ptr, pairing p);
-
-  // MATLAB code generation
-  // classPath is class directory, e.g., ../matlab/@Point2
-
-  void matlab_mfile  (const std::string& classPath); // m-file
-  void matlab_wrapper(const std::string& classPath, 
-		      const std::string& className, 
-		      const std::string& nameSpace); // wrapper
-};
-
diff --git a/wrap/Module.cpp b/wrap/Module.cpp
deleted file mode 100644
index 82977e336..000000000
--- a/wrap/Module.cpp
+++ /dev/null
@@ -1,228 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Module.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-
-//#define BOOST_SPIRIT_DEBUG
-#include <boost/spirit/include/classic_core.hpp>
-#include <boost/foreach.hpp>
-
-#include "Module.h"
-#include "utilities.h"
-
-using namespace std;
-using namespace BOOST_SPIRIT_CLASSIC_NS;
-
-typedef rule<BOOST_SPIRIT_CLASSIC_NS::phrase_scanner_t> Rule;
-
-/* ************************************************************************* */
-// We parse an interface file into a Module object.
-// The grammar is defined using the boost/spirit combinatorial parser.
-// For example, str_p("const") parses the string "const", and the >>
-// operator creates a sequence parser. The grammar below, composed of rules
-// and with start rule [class_p], doubles as the specs for our interface files.
-/* ************************************************************************* */
-
-Module::Module(const string& interfacePath,
-	       const string& moduleName, bool verbose) : name(moduleName), verbose_(verbose)
-{
-  // these variables will be imperatively updated to gradually build [cls]
-  // The one with postfix 0 are used to reset the variables after parse.
-  Argument arg0, arg;
-  ArgumentList args0, args;
-  Constructor constructor0(verbose), constructor(verbose);
-  Method method0(verbose), method(verbose);
-  Class cls0(verbose),cls(verbose);
-
-  //----------------------------------------------------------------------------
-  // Grammar with actions that build the Class object. Actions are
-  // defined within the square brackets [] and are executed whenever a
-  // rule is successfully parsed. Define BOOST_SPIRIT_DEBUG to debug.
-  // The grammar is allows a very restricted C++ header:
-  // - No comments allowed.
-  //  -Only types allowed are string, bool, size_t int, double, Vector, and Matrix
-  //   as well as class names that start with an uppercase letter
-  // - The types unsigned int and bool should be specified as int.
-  // ----------------------------------------------------------------------------
-
-  // lexeme_d turns off white space skipping
-  // http://www.boost.org/doc/libs/1_37_0/libs/spirit/classic/doc/directives.html
-
-  Rule className_p  = lexeme_d[upper_p >> *(alnum_p | '_')];
-
-  Rule classPtr_p =
-    className_p     [assign_a(arg.type)] >> 
-    ch_p('*')       [assign_a(arg.is_ptr,true)];
-
-  Rule classRef_p =
-    !str_p("const") [assign_a(arg.is_const,true)] >> 
-    className_p     [assign_a(arg.type)] >> 
-    ch_p('&')       [assign_a(arg.is_ref,true)];
-
-  Rule basisType_p = 
-    (str_p("string") | "bool" | "size_t" | "int" | "double");
-
-  Rule ublasType = 
-    (str_p("Vector") | "Matrix")[assign_a(arg.type)] >>
-    !ch_p('*')[assign_a(arg.is_ptr,true)];
-
-  Rule name_p = lexeme_d[alpha_p >> *(alnum_p | '_')];
-
-  Rule argument_p = 
-    ((basisType_p[assign_a(arg.type)] | ublasType | classPtr_p | classRef_p) >> name_p[assign_a(arg.name)])
-    [push_back_a(args, arg)]
-    [assign_a(arg,arg0)];
-
-  Rule argumentList_p = !argument_p >> * (',' >> argument_p);
-
-  Rule constructor_p = 
-    (className_p >> '(' >> argumentList_p >> ')' >> ';')
-    [assign_a(constructor.args,args)]
-    [assign_a(args,args0)]
-    [push_back_a(cls.constructors, constructor)]
-    [assign_a(constructor,constructor0)];
-
-  Rule returnType1_p =
-    basisType_p[assign_a(method.returns)] | 
-    ((className_p | "Vector" | "Matrix")[assign_a(method.returns)] >>
-     !ch_p('*')  [assign_a(method.returns_ptr,true)]);
-
-  Rule returnType2_p =
-    basisType_p[assign_a(method.returns2)] | 
-    ((className_p | "Vector" | "Matrix")[assign_a(method.returns2)] >>
-     !ch_p('*')  [assign_a(method.returns_ptr2,true)]);
-
-  Rule pair_p = 
-    (str_p("pair") >> '<' >> returnType1_p >> ',' >> returnType2_p >> '>')
-    [assign_a(method.returns_pair,true)];
-
-  Rule void_p = str_p("void")[assign_a(method.returns)];
-
-  Rule returnType_p = void_p | returnType1_p | pair_p;
-
-  Rule methodName_p = lexeme_d[lower_p >> *(alnum_p | '_')];
-
-  Rule method_p = 
-    (returnType_p >> methodName_p[assign_a(method.name)] >> 
-     '(' >> argumentList_p >> ')' >> 
-     !str_p("const")[assign_a(method.is_const,true)] >> ';')
-    [assign_a(method.args,args)]
-    [assign_a(args,args0)]
-    [push_back_a(cls.methods, method)]
-    [assign_a(method,method0)];
-
-  Rule class_p = str_p("class") >> className_p[assign_a(cls.name)] >> '{' >> 
-    *constructor_p >> 
-    *method_p >> 
-    '}' >> ";";
-
-  Rule module_p = +class_p
-    [push_back_a(classes,cls)]
-    [assign_a(cls,cls0)] 
-    >> !end_p;
-
-  //----------------------------------------------------------------------------
-  // for debugging, define BOOST_SPIRIT_DEBUG
-# ifdef BOOST_SPIRIT_DEBUG
-  BOOST_SPIRIT_DEBUG_NODE(className_p);
-  BOOST_SPIRIT_DEBUG_NODE(classPtr_p);
-  BOOST_SPIRIT_DEBUG_NODE(classRef_p);
-  BOOST_SPIRIT_DEBUG_NODE(basisType_p);
-  BOOST_SPIRIT_DEBUG_NODE(name_p);
-  BOOST_SPIRIT_DEBUG_NODE(argument_p);
-  BOOST_SPIRIT_DEBUG_NODE(argumentList_p);
-  BOOST_SPIRIT_DEBUG_NODE(constructor_p);
-  BOOST_SPIRIT_DEBUG_NODE(returnType1_p);
-  BOOST_SPIRIT_DEBUG_NODE(returnType2_p);
-  BOOST_SPIRIT_DEBUG_NODE(pair_p);
-  BOOST_SPIRIT_DEBUG_NODE(void_p);
-  BOOST_SPIRIT_DEBUG_NODE(returnType_p);
-  BOOST_SPIRIT_DEBUG_NODE(methodName_p);
-  BOOST_SPIRIT_DEBUG_NODE(method_p);
-  BOOST_SPIRIT_DEBUG_NODE(class_p);
-  BOOST_SPIRIT_DEBUG_NODE(module_p);
-# endif
-  //----------------------------------------------------------------------------
-
-  // read interface file
-  string interfaceFile = interfacePath + "/" + moduleName + ".h";
-  string contents = file_contents(interfaceFile);
-
-  // Comment parser : does not work for some reason
-  rule<> comment_p = str_p("/*") >> +anychar_p >> "*/";
-  rule<> skip_p = space_p; // | comment_p;
-
-  // and parse contents
-  parse_info<const char*> info = parse(contents.c_str(), module_p, skip_p); 
-  if(!info.full) {
-    printf("parsing stopped at \n%.20s\n",info.stop);
-    throw ParseFailed(info.length);
-  }
-}
-
-/* ************************************************************************* */
-void Module::matlab_code(const string& toolboxPath, 
-			 const string& nameSpace, 
-			 const string& mexFlags)
-{
-  try {
-    string installCmd = "install -d " + toolboxPath;
-    system(installCmd.c_str());
-
-    // create make m-file
-    string makeFile = toolboxPath + "/make_" + name + ".m";
-    ofstream ofs(makeFile.c_str());
-    if(!ofs) throw CantOpenFile(makeFile);
-
-    if (verbose_) cerr << "generating " << makeFile << endl;
-    emit_header_comment(ofs,"%");
-    ofs << "echo on" << endl << endl;
-    ofs << "toolboxpath = pwd" << endl;
-    ofs << "addpath(toolboxpath);" << endl << endl;
-
-    // generate proxy classes and wrappers
-    BOOST_FOREACH(Class cls, classes) {
-      // create directory if needed
-      string classPath = toolboxPath + "/@" + cls.name;
-      string installCmd = "install -d " + classPath;
-      system(installCmd.c_str());
-
-      // create proxy class
-      string classFile = classPath + "/" + cls.name + ".m";
-      cls.matlab_proxy(classFile);
-
-      // create constructor and method wrappers
-      cls.matlab_constructors(toolboxPath,nameSpace);
-      cls.matlab_methods(classPath,nameSpace);
-
-      // add lines to make m-file
-      ofs << "cd(toolboxpath)" << endl;
-      cls.matlab_make_fragment(ofs, toolboxPath, mexFlags);
-    }  
-
-    // finish make m-file
-    ofs << "cd(toolboxpath)" << endl << endl;
-    ofs << "echo off" << endl;
-    ofs.close();
-  }
-  catch(exception &e) {
-    cerr << "generate_matlab_toolbox failed because " << e.what() << endl;
-  }
-
-}
-
-/* ************************************************************************* */
diff --git a/wrap/Module.h b/wrap/Module.h
deleted file mode 100644
index 5ee1248ca..000000000
--- a/wrap/Module.h
+++ /dev/null
@@ -1,45 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: Module.h
- * brief: describes module to be wrapped
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <string>
-#include <list>
-
-#include "Class.h"
-
-// A module has classes
-struct Module {
-  std::string name;
-  std::list<Class> classes;
-  bool verbose_;
-
-  /**
-   * constructor that parses interface file
-   */
-  Module(const std::string& interfacePath, 
-	 const std::string& moduleName,
-	 bool verbose=true);
-
-  /**
-   *  MATLAB code generation:
-   */
-  void matlab_code(const std::string& path, 
-		   const std::string& nameSpace, 
-		   const std::string& mexFlags);
-};
-
diff --git a/wrap/expected/@Point2/Point2.m b/wrap/expected/@Point2/Point2.m
deleted file mode 100644
index 9cdfd6fd5..000000000
--- a/wrap/expected/@Point2/Point2.m
+++ /dev/null
@@ -1,15 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-classdef Point2
-  properties
-    self = 0
-  end
-  methods
-    function obj = Point2(varargin)
-      if nargin == 0, obj.self = new_Point2_(); end
-      if nargin == 2, obj.self = new_Point2_dd(varargin{1},varargin{2}); end
-      if nargin ~= 13 && obj.self == 0, error('Point2 constructor failed'); end
-    end
-    function display(obj), obj.print(''); end
-    function disp(obj), obj.display; end
-  end
-end
diff --git a/wrap/expected/@Point2/dim.cpp b/wrap/expected/@Point2/dim.cpp
deleted file mode 100644
index 2955d58c3..000000000
--- a/wrap/expected/@Point2/dim.cpp
+++ /dev/null
@@ -1,10 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point2.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("dim",nargout,nargin-1,0);
-  shared_ptr<Point2> self = unwrap_shared_ptr< Point2 >(in[0],"Point2");
-  int result = self->dim();
-  out[0] = wrap< int >(result);
-}
diff --git a/wrap/expected/@Point2/dim.m b/wrap/expected/@Point2/dim.m
deleted file mode 100644
index 93caab12f..000000000
--- a/wrap/expected/@Point2/dim.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.dim()
-function result = dim(obj)
-  error('need to compile dim.cpp');
-end
diff --git a/wrap/expected/@Point2/x.cpp b/wrap/expected/@Point2/x.cpp
deleted file mode 100644
index 5f98d17c6..000000000
--- a/wrap/expected/@Point2/x.cpp
+++ /dev/null
@@ -1,10 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point2.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("x",nargout,nargin-1,0);
-  shared_ptr<Point2> self = unwrap_shared_ptr< Point2 >(in[0],"Point2");
-  double result = self->x();
-  out[0] = wrap< double >(result);
-}
diff --git a/wrap/expected/@Point2/x.m b/wrap/expected/@Point2/x.m
deleted file mode 100644
index 4dae83bf5..000000000
--- a/wrap/expected/@Point2/x.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.x()
-function result = x(obj)
-  error('need to compile x.cpp');
-end
diff --git a/wrap/expected/@Point2/y.cpp b/wrap/expected/@Point2/y.cpp
deleted file mode 100644
index 835aa8a1e..000000000
--- a/wrap/expected/@Point2/y.cpp
+++ /dev/null
@@ -1,10 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point2.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("y",nargout,nargin-1,0);
-  shared_ptr<Point2> self = unwrap_shared_ptr< Point2 >(in[0],"Point2");
-  double result = self->y();
-  out[0] = wrap< double >(result);
-}
diff --git a/wrap/expected/@Point2/y.m b/wrap/expected/@Point2/y.m
deleted file mode 100644
index 9e2897114..000000000
--- a/wrap/expected/@Point2/y.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.y()
-function result = y(obj)
-  error('need to compile y.cpp');
-end
diff --git a/wrap/expected/@Point3/Point3.m b/wrap/expected/@Point3/Point3.m
deleted file mode 100644
index 139c7d079..000000000
--- a/wrap/expected/@Point3/Point3.m
+++ /dev/null
@@ -1,14 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-classdef Point3
-  properties
-    self = 0
-  end
-  methods
-    function obj = Point3(varargin)
-      if nargin == 3, obj.self = new_Point3_ddd(varargin{1},varargin{2},varargin{3}); end
-      if nargin ~= 13 && obj.self == 0, error('Point3 constructor failed'); end
-    end
-    function display(obj), obj.print(''); end
-    function disp(obj), obj.display; end
-  end
-end
diff --git a/wrap/expected/@Point3/norm.cpp b/wrap/expected/@Point3/norm.cpp
deleted file mode 100644
index 37ad329ce..000000000
--- a/wrap/expected/@Point3/norm.cpp
+++ /dev/null
@@ -1,10 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point3.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("norm",nargout,nargin-1,0);
-  shared_ptr<Point3> self = unwrap_shared_ptr< Point3 >(in[0],"Point3");
-  double result = self->norm();
-  out[0] = wrap< double >(result);
-}
diff --git a/wrap/expected/@Point3/norm.m b/wrap/expected/@Point3/norm.m
deleted file mode 100644
index 02943303f..000000000
--- a/wrap/expected/@Point3/norm.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.norm()
-function result = norm(obj)
-  error('need to compile norm.cpp');
-end
diff --git a/wrap/expected/@Test/Test.m b/wrap/expected/@Test/Test.m
deleted file mode 100644
index 31e5e35b2..000000000
--- a/wrap/expected/@Test/Test.m
+++ /dev/null
@@ -1,14 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-classdef Test
-  properties
-    self = 0
-  end
-  methods
-    function obj = Test(varargin)
-      if nargin == 0, obj.self = new_Test_(); end
-      if nargin ~= 13 && obj.self == 0, error('Test constructor failed'); end
-    end
-    function display(obj), obj.print(''); end
-    function disp(obj), obj.display; end
-  end
-end
diff --git a/wrap/expected/@Test/create_ptrs.cpp b/wrap/expected/@Test/create_ptrs.cpp
deleted file mode 100644
index 9a7215a60..000000000
--- a/wrap/expected/@Test/create_ptrs.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("create_ptrs",nargout,nargin-1,0);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  pair< shared_ptr<Test>, shared_ptr<Test> > result = self->create_ptrs();
-  out[0] = wrap_shared_ptr(result.first,"Test");
-  out[1] = wrap_shared_ptr(result.second,"Test");
-}
diff --git a/wrap/expected/@Test/create_ptrs.m b/wrap/expected/@Test/create_ptrs.m
deleted file mode 100644
index 0fbbd9be5..000000000
--- a/wrap/expected/@Test/create_ptrs.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.create_ptrs()
-function [first,second] = create_ptrs(obj)
-  error('need to compile create_ptrs.cpp');
-end
diff --git a/wrap/expected/@Test/print.cpp b/wrap/expected/@Test/print.cpp
deleted file mode 100644
index 5db55477c..000000000
--- a/wrap/expected/@Test/print.cpp
+++ /dev/null
@@ -1,9 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("print",nargout,nargin-1,0);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  self->print();
-}
diff --git a/wrap/expected/@Test/print.m b/wrap/expected/@Test/print.m
deleted file mode 100644
index 91e8ff31a..000000000
--- a/wrap/expected/@Test/print.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.print()
-function result = print(obj)
-  error('need to compile print.cpp');
-end
diff --git a/wrap/expected/@Test/return_TestPtr.cpp b/wrap/expected/@Test/return_TestPtr.cpp
deleted file mode 100644
index 0c22621a6..000000000
--- a/wrap/expected/@Test/return_TestPtr.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_TestPtr",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  shared_ptr<Test> value = unwrap_shared_ptr< Test >(in[1], "Test");
-  shared_ptr<Test> result = self->return_TestPtr(value);
-  out[0] = wrap_shared_ptr(result,"Test");
-}
diff --git a/wrap/expected/@Test/return_TestPtr.m b/wrap/expected/@Test/return_TestPtr.m
deleted file mode 100644
index 0753c283b..000000000
--- a/wrap/expected/@Test/return_TestPtr.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_TestPtr(value)
-function result = return_TestPtr(obj,value)
-  error('need to compile return_TestPtr.cpp');
-end
diff --git a/wrap/expected/@Test/return_bool.cpp b/wrap/expected/@Test/return_bool.cpp
deleted file mode 100644
index d05f5c3b7..000000000
--- a/wrap/expected/@Test/return_bool.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_bool",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  bool value = unwrap< bool >(in[1]);
-  bool result = self->return_bool(value);
-  out[0] = wrap< bool >(result);
-}
diff --git a/wrap/expected/@Test/return_bool.m b/wrap/expected/@Test/return_bool.m
deleted file mode 100644
index 3ef97ad8f..000000000
--- a/wrap/expected/@Test/return_bool.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_bool(value)
-function result = return_bool(obj,value)
-  error('need to compile return_bool.cpp');
-end
diff --git a/wrap/expected/@Test/return_double.cpp b/wrap/expected/@Test/return_double.cpp
deleted file mode 100644
index edbffb889..000000000
--- a/wrap/expected/@Test/return_double.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_double",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  double value = unwrap< double >(in[1]);
-  double result = self->return_double(value);
-  out[0] = wrap< double >(result);
-}
diff --git a/wrap/expected/@Test/return_double.m b/wrap/expected/@Test/return_double.m
deleted file mode 100644
index 87e2929da..000000000
--- a/wrap/expected/@Test/return_double.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_double(value)
-function result = return_double(obj,value)
-  error('need to compile return_double.cpp');
-end
diff --git a/wrap/expected/@Test/return_field.cpp b/wrap/expected/@Test/return_field.cpp
deleted file mode 100644
index 727f6faa0..000000000
--- a/wrap/expected/@Test/return_field.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_field",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Test& t = *unwrap_shared_ptr< Test >(in[1], "Test");
-  bool result = self->return_field(t);
-  out[0] = wrap< bool >(result);
-}
diff --git a/wrap/expected/@Test/return_field.m b/wrap/expected/@Test/return_field.m
deleted file mode 100644
index 7e63e32e2..000000000
--- a/wrap/expected/@Test/return_field.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_field(t)
-function result = return_field(obj,t)
-  error('need to compile return_field.cpp');
-end
diff --git a/wrap/expected/@Test/return_int.cpp b/wrap/expected/@Test/return_int.cpp
deleted file mode 100644
index 61b53f4ce..000000000
--- a/wrap/expected/@Test/return_int.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_int",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  int value = unwrap< int >(in[1]);
-  int result = self->return_int(value);
-  out[0] = wrap< int >(result);
-}
diff --git a/wrap/expected/@Test/return_int.m b/wrap/expected/@Test/return_int.m
deleted file mode 100644
index df5502203..000000000
--- a/wrap/expected/@Test/return_int.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_int(value)
-function result = return_int(obj,value)
-  error('need to compile return_int.cpp');
-end
diff --git a/wrap/expected/@Test/return_matrix1.cpp b/wrap/expected/@Test/return_matrix1.cpp
deleted file mode 100644
index c7b12ddad..000000000
--- a/wrap/expected/@Test/return_matrix1.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_matrix1",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Matrix value = unwrap< Matrix >(in[1]);
-  Matrix result = self->return_matrix1(value);
-  out[0] = wrap< Matrix >(result);
-}
diff --git a/wrap/expected/@Test/return_matrix1.m b/wrap/expected/@Test/return_matrix1.m
deleted file mode 100644
index bb71670db..000000000
--- a/wrap/expected/@Test/return_matrix1.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_matrix1(value)
-function result = return_matrix1(obj,value)
-  error('need to compile return_matrix1.cpp');
-end
diff --git a/wrap/expected/@Test/return_matrix2.cpp b/wrap/expected/@Test/return_matrix2.cpp
deleted file mode 100644
index e41cf5f96..000000000
--- a/wrap/expected/@Test/return_matrix2.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_matrix2",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Matrix value = unwrap< Matrix >(in[1]);
-  Matrix result = self->return_matrix2(value);
-  out[0] = wrap< Matrix >(result);
-}
diff --git a/wrap/expected/@Test/return_matrix2.m b/wrap/expected/@Test/return_matrix2.m
deleted file mode 100644
index b8b74dd03..000000000
--- a/wrap/expected/@Test/return_matrix2.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_matrix2(value)
-function result = return_matrix2(obj,value)
-  error('need to compile return_matrix2.cpp');
-end
diff --git a/wrap/expected/@Test/return_pair.cpp b/wrap/expected/@Test/return_pair.cpp
deleted file mode 100644
index 0e274049e..000000000
--- a/wrap/expected/@Test/return_pair.cpp
+++ /dev/null
@@ -1,13 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_pair",nargout,nargin-1,2);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Vector v = unwrap< Vector >(in[1]);
-  Matrix A = unwrap< Matrix >(in[2]);
-  pair< Vector, Matrix > result = self->return_pair(v,A);
-  out[0] = wrap< Vector >(result.first);
-  out[1] = wrap< Matrix >(result.second);
-}
diff --git a/wrap/expected/@Test/return_pair.m b/wrap/expected/@Test/return_pair.m
deleted file mode 100644
index 35348fd9b..000000000
--- a/wrap/expected/@Test/return_pair.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_pair(v,A)
-function [first,second] = return_pair(obj,v,A)
-  error('need to compile return_pair.cpp');
-end
diff --git a/wrap/expected/@Test/return_ptrs.cpp b/wrap/expected/@Test/return_ptrs.cpp
deleted file mode 100644
index 4ad4c4fd4..000000000
--- a/wrap/expected/@Test/return_ptrs.cpp
+++ /dev/null
@@ -1,13 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_ptrs",nargout,nargin-1,2);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  shared_ptr<Test> p1 = unwrap_shared_ptr< Test >(in[1], "Test");
-  shared_ptr<Test> p2 = unwrap_shared_ptr< Test >(in[2], "Test");
-  pair< shared_ptr<Test>, shared_ptr<Test> > result = self->return_ptrs(p1,p2);
-  out[0] = wrap_shared_ptr(result.first,"Test");
-  out[1] = wrap_shared_ptr(result.second,"Test");
-}
diff --git a/wrap/expected/@Test/return_ptrs.m b/wrap/expected/@Test/return_ptrs.m
deleted file mode 100644
index ba1a16659..000000000
--- a/wrap/expected/@Test/return_ptrs.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_ptrs(p1,p2)
-function [first,second] = return_ptrs(obj,p1,p2)
-  error('need to compile return_ptrs.cpp');
-end
diff --git a/wrap/expected/@Test/return_size_t.cpp b/wrap/expected/@Test/return_size_t.cpp
deleted file mode 100644
index ce5a1a9e4..000000000
--- a/wrap/expected/@Test/return_size_t.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_size_t",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  size_t value = unwrap< size_t >(in[1]);
-  size_t result = self->return_size_t(value);
-  out[0] = wrap< size_t >(result);
-}
diff --git a/wrap/expected/@Test/return_size_t.m b/wrap/expected/@Test/return_size_t.m
deleted file mode 100644
index ad4500fd1..000000000
--- a/wrap/expected/@Test/return_size_t.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_size_t(value)
-function result = return_size_t(obj,value)
-  error('need to compile return_size_t.cpp');
-end
diff --git a/wrap/expected/@Test/return_string.cpp b/wrap/expected/@Test/return_string.cpp
deleted file mode 100644
index d09e8fb31..000000000
--- a/wrap/expected/@Test/return_string.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_string",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  string value = unwrap< string >(in[1]);
-  string result = self->return_string(value);
-  out[0] = wrap< string >(result);
-}
diff --git a/wrap/expected/@Test/return_string.m b/wrap/expected/@Test/return_string.m
deleted file mode 100644
index 28a02b0d1..000000000
--- a/wrap/expected/@Test/return_string.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_string(value)
-function result = return_string(obj,value)
-  error('need to compile return_string.cpp');
-end
diff --git a/wrap/expected/@Test/return_vector1.cpp b/wrap/expected/@Test/return_vector1.cpp
deleted file mode 100644
index 079ebff09..000000000
--- a/wrap/expected/@Test/return_vector1.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_vector1",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Vector value = unwrap< Vector >(in[1]);
-  Vector result = self->return_vector1(value);
-  out[0] = wrap< Vector >(result);
-}
diff --git a/wrap/expected/@Test/return_vector1.m b/wrap/expected/@Test/return_vector1.m
deleted file mode 100644
index 266017c86..000000000
--- a/wrap/expected/@Test/return_vector1.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_vector1(value)
-function result = return_vector1(obj,value)
-  error('need to compile return_vector1.cpp');
-end
diff --git a/wrap/expected/@Test/return_vector2.cpp b/wrap/expected/@Test/return_vector2.cpp
deleted file mode 100644
index 693d3d06b..000000000
--- a/wrap/expected/@Test/return_vector2.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("return_vector2",nargout,nargin-1,1);
-  shared_ptr<Test> self = unwrap_shared_ptr< Test >(in[0],"Test");
-  Vector value = unwrap< Vector >(in[1]);
-  Vector result = self->return_vector2(value);
-  out[0] = wrap< Vector >(result);
-}
diff --git a/wrap/expected/@Test/return_vector2.m b/wrap/expected/@Test/return_vector2.m
deleted file mode 100644
index f237ce228..000000000
--- a/wrap/expected/@Test/return_vector2.m
+++ /dev/null
@@ -1,5 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-% usage: obj.return_vector2(value)
-function result = return_vector2(obj,value)
-  error('need to compile return_vector2.cpp');
-end
diff --git a/wrap/expected/make_geometry.m b/wrap/expected/make_geometry.m
deleted file mode 100644
index a266b2d6d..000000000
--- a/wrap/expected/make_geometry.m
+++ /dev/null
@@ -1,44 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-echo on
-
-toolboxpath = pwd
-addpath(toolboxpath);
-
-cd(toolboxpath)
-mex -O5 new_Point2_.cpp
-mex -O5 new_Point2_dd.cpp
-
-cd @Point2
-mex -O5 x.cpp
-mex -O5 y.cpp
-mex -O5 dim.cpp
-
-cd(toolboxpath)
-mex -O5 new_Point3_ddd.cpp
-
-cd @Point3
-mex -O5 norm.cpp
-
-cd(toolboxpath)
-mex -O5 new_Test_.cpp
-
-cd @Test
-mex -O5 return_bool.cpp
-mex -O5 return_size_t.cpp
-mex -O5 return_int.cpp
-mex -O5 return_double.cpp
-mex -O5 return_string.cpp
-mex -O5 return_vector1.cpp
-mex -O5 return_matrix1.cpp
-mex -O5 return_vector2.cpp
-mex -O5 return_matrix2.cpp
-mex -O5 return_pair.cpp
-mex -O5 return_field.cpp
-mex -O5 return_TestPtr.cpp
-mex -O5 create_ptrs.cpp
-mex -O5 return_ptrs.cpp
-mex -O5 print.cpp
-
-cd(toolboxpath)
-
-echo off
diff --git a/wrap/expected/new_Point2_.cpp b/wrap/expected/new_Point2_.cpp
deleted file mode 100644
index 6a78d318d..000000000
--- a/wrap/expected/new_Point2_.cpp
+++ /dev/null
@@ -1,9 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point2.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("new_Point2_",nargout,nargin,0);
-  Point2* self = new Point2();
-  out[0] = wrap_constructed(self,"Point2");
-}
diff --git a/wrap/expected/new_Point2_.m b/wrap/expected/new_Point2_.m
deleted file mode 100644
index b5cab71ab..000000000
--- a/wrap/expected/new_Point2_.m
+++ /dev/null
@@ -1,4 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-function result = new_Point2_(obj)
-  error('need to compile new_Point2_.cpp');
-end
diff --git a/wrap/expected/new_Point2_dd.cpp b/wrap/expected/new_Point2_dd.cpp
deleted file mode 100644
index 15a8a190d..000000000
--- a/wrap/expected/new_Point2_dd.cpp
+++ /dev/null
@@ -1,11 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point2.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("new_Point2_dd",nargout,nargin,2);
-  double x = unwrap< double >(in[0]);
-  double y = unwrap< double >(in[1]);
-  Point2* self = new Point2(x,y);
-  out[0] = wrap_constructed(self,"Point2");
-}
diff --git a/wrap/expected/new_Point2_dd.m b/wrap/expected/new_Point2_dd.m
deleted file mode 100644
index bdd42b766..000000000
--- a/wrap/expected/new_Point2_dd.m
+++ /dev/null
@@ -1,4 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-function result = new_Point2_dd(obj,x,y)
-  error('need to compile new_Point2_dd.cpp');
-end
diff --git a/wrap/expected/new_Point3_ddd.cpp b/wrap/expected/new_Point3_ddd.cpp
deleted file mode 100644
index 82c95bdc6..000000000
--- a/wrap/expected/new_Point3_ddd.cpp
+++ /dev/null
@@ -1,12 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Point3.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("new_Point3_ddd",nargout,nargin,3);
-  double x = unwrap< double >(in[0]);
-  double y = unwrap< double >(in[1]);
-  double z = unwrap< double >(in[2]);
-  Point3* self = new Point3(x,y,z);
-  out[0] = wrap_constructed(self,"Point3");
-}
diff --git a/wrap/expected/new_Point3_ddd.m b/wrap/expected/new_Point3_ddd.m
deleted file mode 100644
index 229b9658c..000000000
--- a/wrap/expected/new_Point3_ddd.m
+++ /dev/null
@@ -1,4 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-function result = new_Point3_ddd(obj,x,y,z)
-  error('need to compile new_Point3_ddd.cpp');
-end
diff --git a/wrap/expected/new_Test_.cpp b/wrap/expected/new_Test_.cpp
deleted file mode 100644
index ab2171762..000000000
--- a/wrap/expected/new_Test_.cpp
+++ /dev/null
@@ -1,9 +0,0 @@
-// automatically generated by wrap on 2010-Feb-23
-#include <wrap/matlab.h>
-#include <Test.h>
-void mexFunction(int nargout, mxArray *out[], int nargin, const mxArray *in[])
-{
-  checkArguments("new_Test_",nargout,nargin,0);
-  Test* self = new Test();
-  out[0] = wrap_constructed(self,"Test");
-}
diff --git a/wrap/expected/new_Test_.m b/wrap/expected/new_Test_.m
deleted file mode 100644
index 1aae9b6f5..000000000
--- a/wrap/expected/new_Test_.m
+++ /dev/null
@@ -1,4 +0,0 @@
-% automatically generated by wrap on 2010-Feb-23
-function result = new_Test_(obj)
-  error('need to compile new_Test_.cpp');
-end
diff --git a/wrap/geometry.h b/wrap/geometry.h
deleted file mode 100644
index 9b1d80b00..000000000
--- a/wrap/geometry.h
+++ /dev/null
@@ -1,39 +0,0 @@
-
-class Point2 {
- Point2();
- Point2(double x, double y);
- double x();
- double y();
- int dim() const;
-};
-
-class Point3 {
-  Point3(double x, double y, double z);
-  double norm() const;
-};
-
-class Test {
-  Test();
-
-  bool   return_bool   (bool   value);
-  size_t return_size_t (size_t value);
-  int    return_int    (int    value);
-  double return_double (double value);
-
-  string return_string (string value);
-  Vector return_vector1(Vector value);
-  Matrix return_matrix1(Matrix value);
-  Vector return_vector2(Vector value);
-  Matrix return_matrix2(Matrix value);
-
-  pair<Vector,Matrix> return_pair (Vector v, Matrix A);
-
-  bool return_field(const Test& t) const;
-
-  Test* return_TestPtr(Test* value);
-
-  pair<Test*,Test*> create_ptrs ();
-  pair<Test*,Test*> return_ptrs (Test* p1, Test* p2);
-
-  void print();
-};
diff --git a/wrap/hello.cpp b/wrap/hello.cpp
deleted file mode 100644
index a140afd11..000000000
--- a/wrap/hello.cpp
+++ /dev/null
@@ -1,26 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-// example for wrapping python with boost
-// from http://www.boost.org/doc/libs/1_37_0/libs/python/doc/tutorial/doc/html/index.html
-
-char const* greet()
-{
-   return "hello, world";
-}
-
-#include <boost/python.hpp>
-
-BOOST_PYTHON_MODULE(hello_ext)
-{
-    using namespace boost::python;
-    def("greet", greet);
-}
diff --git a/wrap/testSpirit.cpp b/wrap/testSpirit.cpp
deleted file mode 100644
index 120e5561f..000000000
--- a/wrap/testSpirit.cpp
+++ /dev/null
@@ -1,105 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * Unit test for Boost's awesome Spirit parser
- * Author: Frank Dellaert
- **/
-
-#include <boost/spirit/include/classic_core.hpp>
-#include <boost/spirit/include/classic_push_back_actor.hpp>
-#include <CppUnitLite/TestHarness.h>
-
-using namespace std;
-using namespace BOOST_SPIRIT_CLASSIC_NS;
-
-typedef rule<BOOST_SPIRIT_CLASSIC_NS::phrase_scanner_t> Rule;
-
-/* ************************************************************************* */
-// lexeme_d turns off white space skipping
-// http://www.boost.org/doc/libs/1_37_0/libs/spirit/classic/doc/directives.html
-Rule name_p       = lexeme_d[alpha_p >> *(alnum_p | '_')];
-Rule className_p  = lexeme_d[upper_p >> *(alnum_p | '_')];
-Rule methodName_p = lexeme_d[lower_p >> *(alnum_p | '_')];
-
-Rule basisType_p = (str_p("string") | "bool" | "size_t" | "int" | "double" | "Vector" | "Matrix");
-
-/* ************************************************************************* */
-TEST( spirit, real ) {
-  // check if we can parse 8.99 as a real
-  CHECK(parse("8.99", real_p, space_p).full);
-  // make sure parsing fails on this one
-  CHECK(!parse("zztop", real_p, space_p).full);
-}
-
-/* ************************************************************************* */
-TEST( spirit, string ) {
-  // check if we can parse a string
-  CHECK(parse("double", str_p("double"), space_p).full);
-}
-
-/* ************************************************************************* */
-TEST( spirit, sequence ) {
-  // check that we skip white space
-  CHECK(parse("int int", str_p("int") >> *str_p("int"), space_p).full);
-  CHECK(parse("int --- - -- -", str_p("int") >> *ch_p('-'), space_p).full);
-  CHECK(parse("const \t string", str_p("const") >> str_p("string"), space_p).full);
-
-  // not that (see spirit FAQ) the vanilla rule<> does not deal with whitespace
-  rule<>vanilla_p = str_p("const") >> str_p("string");
-  CHECK(!parse("const \t string", vanilla_p, space_p).full);
-
-  // to fix it, we need to use <phrase_scanner_t>
-  rule<phrase_scanner_t>phrase_level_p = str_p("const") >> str_p("string");
-  CHECK(parse("const \t string", phrase_level_p, space_p).full);
-}
-
-/* ************************************************************************* */
-// parser for interface files
-
-// const string reference reference
-Rule constStringRef_p = 
-  str_p("const") >> "string" >> '&';
-
-// class reference
-Rule classRef_p = className_p >> '&';
-
-// const class reference
-Rule constClassRef_p = str_p("const") >> classRef_p;
-
-// method parsers
-Rule constMethod_p = basisType_p >> methodName_p >> '(' >> ')' >> "const" >> ';';
-
-/* ************************************************************************* */
-TEST( spirit, basisType_p ) {
-  CHECK(!parse("Point3", basisType_p, space_p).full);
-  CHECK(parse("string", basisType_p, space_p).full);
-}
-
-/* ************************************************************************* */
-TEST( spirit, className_p ) {
-  CHECK(parse("Point3", className_p, space_p).full);
-}
-
-/* ************************************************************************* */
-TEST( spirit, classRef_p ) {
-  CHECK(parse("Point3 &", classRef_p, space_p).full);
-  CHECK(parse("Point3&", classRef_p, space_p).full);
-}
-
-/* ************************************************************************* */
-TEST( spirit, constMethod_p ) {
-  CHECK(parse("double norm() const;", constMethod_p, space_p).full);
-}
-
-/* ************************************************************************* */
-int main() { TestResult tr; return TestRegistry::runAllTests(tr); }
-/* ************************************************************************* */
diff --git a/wrap/testWrap.cpp b/wrap/testWrap.cpp
deleted file mode 100644
index 6e9f84d59..000000000
--- a/wrap/testWrap.cpp
+++ /dev/null
@@ -1,117 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * Unit test for wrap.c
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-#include <sstream>
-#include <CppUnitLite/TestHarness.h>
-
-#include "utilities.h"
-#include "Module.h"
-
-using namespace std;
-static bool verbose = false;
-#ifdef TOPSRCDIR
-static string topdir = TOPSRCDIR;
-#else
-static string topdir = "..";
-#endif
-
-/* ************************************************************************* */
-TEST( wrap, ArgumentList ) {
-  ArgumentList args;
-  Argument arg; arg.type = "double"; arg.name = "x";
-  args.push_back(arg);
-  args.push_back(arg);
-  args.push_back(arg);
-  CHECK(args.signature()=="ddd");
-  CHECK(args.types()=="double,double,double");
-  CHECK(args.names()=="x,x,x");
-}
-
-/* ************************************************************************* */
-TEST( wrap, check_exception ) {
-	THROWS_EXCEPTION(Module("/notarealpath", "geometry",verbose));
-	CHECK_EXCEPTION(Module("/alsonotarealpath", "geometry",verbose), CantOpenFile);
-}
-
-/* ************************************************************************* */
-TEST( wrap, parse ) {
-	string path = topdir + "/wrap";
-
-	Module module(path.c_str(), "geometry",verbose);
-	CHECK(module.classes.size()==3);
-
-	// check second class, Point3
-	Class cls = *(++module.classes.begin());
-	CHECK(cls.name=="Point3");
-	CHECK(cls.constructors.size()==1);
-	CHECK(cls.methods.size()==1);
-
-	// first constructor takes 3 doubles
-	Constructor c1 = cls.constructors.front();
-	CHECK(c1.args.size()==3);
-
-	// check first double argument
-	Argument a1 = c1.args.front();
-	CHECK(!a1.is_const);
-	CHECK(a1.type=="double");
-	CHECK(!a1.is_ref);
-	CHECK(a1.name=="x");
-
-	// check method
-	Method m1 = cls.methods.front();
-	CHECK(m1.returns=="double");
-	CHECK(m1.name=="norm");
-	CHECK(m1.args.size()==0);
-	CHECK(m1.is_const);
-}
-
-/* ************************************************************************* */
-TEST( wrap, matlab_code ) {
-	// Parse into class object
-	string path = topdir + "/wrap";
-	Module module(path,"geometry",verbose);
-
-	// emit MATLAB code
-	// make_geometry will not compile, use make testwrap to generate real make
-	module.matlab_code("actual", "", "-O5");
-
-	CHECK(files_equal(path + "/expected/@Point2/Point2.m"  , "actual/@Point2/Point2.m"  ));
-	CHECK(files_equal(path + "/expected/@Point2/x.cpp"     , "actual/@Point2/x.cpp"     ));
-
-	CHECK(files_equal(path + "/expected/@Point3/Point3.m"  , "actual/@Point3/Point3.m"  ));
-	CHECK(files_equal(path + "/expected/new_Point3_ddd.m"  , "actual/new_Point3_ddd.m"  ));
-	CHECK(files_equal(path + "/expected/new_Point3_ddd.cpp", "actual/new_Point3_ddd.cpp"));
-	CHECK(files_equal(path + "/expected/@Point3/norm.m"    , "actual/@Point3/norm.m"    ));
-	CHECK(files_equal(path + "/expected/@Point3/norm.cpp"  , "actual/@Point3/norm.cpp"  ));
-
-	CHECK(files_equal(path + "/expected/new_Test_.cpp"           , "actual/new_Test_.cpp"           ));
-	CHECK(files_equal(path + "/expected/@Test/Test.m"            , "actual/@Test/Test.m"            ));
-	CHECK(files_equal(path + "/expected/@Test/return_string.cpp" , "actual/@Test/return_string.cpp" ));
-	CHECK(files_equal(path + "/expected/@Test/return_pair.cpp"   , "actual/@Test/return_pair.cpp"   ));
-	CHECK(files_equal(path + "/expected/@Test/return_field.cpp"  , "actual/@Test/return_field.cpp"  ));
-	CHECK(files_equal(path + "/expected/@Test/return_TestPtr.cpp", "actual/@Test/return_TestPtr.cpp"));
-	CHECK(files_equal(path + "/expected/@Test/return_ptrs.cpp"   , "actual/@Test/return_ptrs.cpp"   ));
-	CHECK(files_equal(path + "/expected/@Test/print.m"           , "actual/@Test/print.m"           ));
-	CHECK(files_equal(path + "/expected/@Test/print.cpp"         , "actual/@Test/print.cpp"         ));
-
-	CHECK(files_equal(path + "/expected/make_geometry.m"   , "actual/make_geometry.m"   ));
-}
-
-/* ************************************************************************* */
-int main() { TestResult tr; return TestRegistry::runAllTests(tr); }
-/* ************************************************************************* */
diff --git a/wrap/utilities.cpp b/wrap/utilities.cpp
deleted file mode 100644
index f7376390f..000000000
--- a/wrap/utilities.cpp
+++ /dev/null
@@ -1,68 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: utilities.ccp
- * Author: Frank Dellaert
- **/
-
-#include <iostream>
-#include <fstream>
-
-#include <boost/date_time/gregorian/gregorian.hpp>
-
-#include "utilities.h"
-
-using namespace std;
-using namespace boost::gregorian;
-
-/* ************************************************************************* */
-string file_contents(const string& filename, bool skipheader) {
-  ifstream ifs(filename.c_str());
-  if(!ifs) throw CantOpenFile(filename);
-
-  // read file into stringstream
-  stringstream ss;
-  if (skipheader) ifs.ignore(256,'\n');
-  ss << ifs.rdbuf();
-  ifs.close();
-
-  // return string
-  return ss.str();
-}
-
-/* ************************************************************************* */
-bool files_equal(const string& expected, const string& actual, bool skipheader) {
-  try {
-    string expected_contents = file_contents(expected, skipheader);
-    string actual_contents   = file_contents(actual, skipheader);
-    bool equal = actual_contents == expected_contents;
-    if (!equal) {
-      stringstream command;
-      command << "diff " << actual << " " << expected << endl;
-      system(command.str().c_str());
-    }
-    return equal;
-  }
-  catch (const string& reason) {
-    cerr << "expection: " << reason << endl;
-    return false;
-  }
-  return true;
-}
-
-/* ************************************************************************* */
-void emit_header_comment(ofstream& ofs, const string& delimiter) {
-  date today = day_clock::local_day();
-  ofs << delimiter << " automatically generated by wrap on " << today << endl;
-}
-
-/* ************************************************************************* */
diff --git a/wrap/utilities.h b/wrap/utilities.h
deleted file mode 100644
index e75c44025..000000000
--- a/wrap/utilities.h
+++ /dev/null
@@ -1,60 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: utilities.ccp
- * Author: Frank Dellaert
- **/
-
-#pragma once
-
-#include <exception>
-#include <sstream>
-
-class CantOpenFile : public std::exception {
- private:
-  std::string filename_;
- public:
- CantOpenFile(const std::string& filename) : filename_(filename) {}
-  ~CantOpenFile() throw() {}
-  virtual const char* what() const throw() { 
-    return ("Can't open file " + filename_).c_str(); 
-  }
-};
-
-class ParseFailed : public std::exception {
- private:
-  int length_;
- public:
- ParseFailed(int length) : length_(length) {}
-  ~ParseFailed() throw() {}
-  virtual const char* what() const throw() { 
-    std::stringstream buf;
-    buf << "Parse failed at character " << (length_+1);
-    return buf.str().c_str(); 
-  }
-};
-
-/**
- * read contents of a file into a std::string
- */
-std::string file_contents(const std::string& filename, bool skipheader=false);
-
-/**
- * Check whether two files are equal
- * By default, skips the first line of actual so header is not generated
- */
-bool files_equal(const std::string& expected, const std::string& actual, bool skipheader=true);
-
-/**
- * emit a header at the top of generated files
- */
-void emit_header_comment(std::ofstream& ofs, const std::string& delimiter);
diff --git a/wrap/wrap-matlab.h b/wrap/wrap-matlab.h
deleted file mode 100644
index 6889daaa6..000000000
--- a/wrap/wrap-matlab.h
+++ /dev/null
@@ -1,438 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-// header file to be included in MATLAB wrappers
-// Copyright (c) 2008 Frank Dellaert, All Rights reserved
-// wrapping and unwrapping is done using specialized templates, see
-// http://www.cplusplus.com/doc/tutorial/templates.html
-
-extern "C" {
-#include <mex.h>
-}
-
-#include <list>
-#include <string>
-#include <sstream>
-#include <boost/numeric/ublas/vector.hpp>
-#include <boost/numeric/ublas/matrix.hpp>
-#include <boost/shared_ptr.hpp>
-
-using namespace std;
-using namespace boost; // not usual, but for conciseness of generated code
-
-// start GTSAM Specifics /////////////////////////////////////////////////
-typedef numeric::ublas::vector<double> Vector;
-typedef numeric::ublas::matrix<double> Matrix;
-// to make keys be constructed from strings:
-#define GTSAM_MAGIC_KEY
-// to enable Matrix and Vector constructor for SharedGaussian:
-#define GTSAM_MAGIC_GAUSSIAN
-// end GTSAM Specifics /////////////////////////////////////////////////
-
-#ifdef __LP64__
-// 64-bit Mac
-#define mxUINT32OR64_CLASS mxUINT64_CLASS
-#else
-#define mxUINT32OR64_CLASS mxUINT32_CLASS
-#endif
-
-//*****************************************************************************
-// Utilities
-//*****************************************************************************
-
-void error(const char* str) {
-  mexErrMsgIdAndTxt("wrap:error", str);
-}
-
-mxArray *scalar(mxClassID classid) {
-  mwSize dims[1]; dims[0]=1;
-  return mxCreateNumericArray(1, dims, classid, mxREAL);
-}
-
-void checkScalar(const mxArray* array, const char* str) {
-	int m = mxGetM(array), n = mxGetN(array);
-	if (m!=1 || n!=1)
-		mexErrMsgIdAndTxt("wrap: not a scalar in ", str);
-}
-
-//*****************************************************************************
-// Check arguments
-//*****************************************************************************
-
-void checkArguments(const string& name, int nargout, int nargin, int expected) {
-  stringstream err; 
-  err << name << " expects " << expected << " arguments, not " << nargin;
-  if (nargin!=expected)
-    error(err.str().c_str());
-}
-
-//*****************************************************************************
-// wrapping C++ basis types in MATLAB arrays
-//*****************************************************************************
-
-// default wrapping throws an error: only basis types are allowed in wrap
-template <typename Class>
-mxArray* wrap(Class& value) {
-  error("wrap internal error: attempted wrap of invalid type");
-}
-
-// specialization to string
-// wraps into a character array
-template<>
-mxArray* wrap<string>(string& value) {
-  return mxCreateString(value.c_str());
-}
-
-// specialization to bool
-template<>
-mxArray* wrap<bool>(bool& value) {
-  mxArray *result = scalar(mxUINT32OR64_CLASS);
-  *(bool*)mxGetData(result) = value;
-  return result;
-}
-
-// specialization to size_t
-template<>
-mxArray* wrap<size_t>(size_t& value) {
-  mxArray *result = scalar(mxUINT32OR64_CLASS);
-  *(size_t*)mxGetData(result) = value;
-  return result;
-}
-
-// specialization to int
-template<>
-mxArray* wrap<int>(int& value) {
-  mxArray *result = scalar(mxUINT32OR64_CLASS);
-  *(int*)mxGetData(result) = value;
-  return result;
-}
-
-// specialization to double -> just double
-template<>
-mxArray* wrap<double>(double& value) {
-  return mxCreateDoubleScalar(value);
-}
-
-// wrap a const BOOST vector into a double vector
-mxArray* wrap_Vector(const Vector& v) {
-  int m = v.size();
-  mxArray *result = mxCreateDoubleMatrix(m, 1, mxREAL);
-  double *data = mxGetPr(result);
-  for (int i=0;i<m;i++) data[i]=v(i);
-  return result;
-}
-
-// specialization to BOOST vector -> double vector
-template<>
-mxArray* wrap<Vector >(Vector& v) {
-  return wrap_Vector(v);
-}
-
-// const version
-template<>
-mxArray* wrap<const Vector >(const Vector& v) {
-  return wrap_Vector(v);
-}
-
-// wrap a const BOOST MATRIX into a double matrix
-mxArray* wrap_Matrix(const Matrix& A) {
-  int m = A.size1(), n = A.size2();
-  mxArray *result = mxCreateDoubleMatrix(m, n, mxREAL);
-  double *data = mxGetPr(result);
-  // converts from column-major to row-major
-  for (int j=0;j<n;j++) for (int i=0;i<m;i++,data++) *data = A(i,j);
-  return result;
-}
-
-// specialization to BOOST MATRIX -> double matrix
-template<>
-mxArray* wrap<Matrix >(Matrix& A) {
-  return wrap_Matrix(A);
-}
-
-// const version
-template<>
-mxArray* wrap<const Matrix >(const Matrix& A) {
-  return wrap_Matrix(A);
-}
-
-//*****************************************************************************
-// unwrapping MATLAB arrays into C++ basis types
-//*****************************************************************************
-
-// default unwrapping throws an error
-// as wrap only supports passing a reference or one of the basic types
-template <typename T>
-T unwrap(const mxArray* array) {
-  error("wrap internal error: attempted unwrap of invalid type");
-}
-
-// specialization to string
-// expects a character array
-// Warning: relies on mxChar==char
-template<>
-string unwrap<string>(const mxArray* array) {
-  char *data = mxArrayToString(array);
-  if (data==NULL) error("unwrap<string>: not a character array");
-  string str(data);
-  mxFree(data);
-  return str;
-}
-
-// specialization to bool
-template<>
-bool unwrap<bool>(const mxArray* array) {
-	checkScalar(array,"unwrap<bool>");
-  return mxGetScalar(array) != 0.0;
-}
-
-// specialization to size_t
-template<>
-size_t unwrap<size_t>(const mxArray* array) {
-	checkScalar(array,"unwrap<size_t>");
-  return (size_t)mxGetScalar(array);
-}
-
-// specialization to int
-template<>
-int unwrap<int>(const mxArray* array) {
-	checkScalar(array,"unwrap<int>");
-  return (int)mxGetScalar(array);
-}
-
-// specialization to double
-template<>
-double unwrap<double>(const mxArray* array) {
-	checkScalar(array,"unwrap<double>");
-  return (double)mxGetScalar(array);
-}
-
-// specialization to BOOST vector
-template<>
-Vector unwrap< Vector >(const mxArray* array) {
-  int m = mxGetM(array), n = mxGetN(array);
-  if (mxIsDouble(array)==false || n!=1) error("unwrap<vector>: not a vector");
-  double* data = (double*)mxGetData(array);
-  Vector v(m);
-  copy(data,data+m,v.begin());
-  return v;
-}
-
-// specialization to BOOST matrix
-template<>
-Matrix unwrap< Matrix >(const mxArray* array) {
-  if (mxIsDouble(array)==false) error("unwrap<matrix>: not a matrix");
-  int m = mxGetM(array), n = mxGetN(array);
-  double* data = (double*)mxGetData(array);
-  Matrix A(m,n);
-  // converts from row-major to column-major
-  for (int j=0;j<n;j++) for (int i=0;i<m;i++,data++) A(i,j) = *data;
-  return A;
-}
-
-//*****************************************************************************
-// Shared Pointer Handle
-// inspired by mexhandle, but using shared_ptr
-//*****************************************************************************
-
-template<typename T> class Collector;
-
-template<typename T>
-class ObjectHandle {
-private:
-	ObjectHandle* signature; // use 'this' as a unique object signature
-	const std::type_info* type; // type checking information
-	shared_ptr<T> t; // object pointer
-
-public:
-	// Constructor for free-store allocated objects.
-	// Creates shared pointer, will delete if is last one to hold pointer
-	ObjectHandle(T* ptr) :
-		type(&typeid(T)), t(shared_ptr<T> (ptr)) {
-		signature = this;
-		Collector<T>::register_handle(this);
-	}
-
-	// Constructor for shared pointers
-	// Creates shared pointer, will delete if is last one to hold pointer
-	ObjectHandle(shared_ptr<T> ptr) :
-		type(&typeid(T)), t(ptr) {
-		signature = this;
-	}
-
-	~ObjectHandle() {
-		// object is in shared_ptr, will be automatically deleted
-		signature = 0; // destroy signature
-	}
-
-	// Get the actual object contained by handle
-	shared_ptr<T> get_object() const {
-		return t;
-	}
-
-	// Print the mexhandle for debugging
-	void print(const char* str) {
-		mexPrintf("mexhandle %s:\n", str);
-		mexPrintf("  signature = %d:\n", signature);
-		mexPrintf("  pointer   = %d:\n", t.get());
-	}
-
-	// Convert ObjectHandle<T> to a mxArray handle (to pass back from mex-function).
-	// Create a numeric array as handle for an ObjectHandle.
-	// We ASSUME we can store object pointer in the mxUINT32 element of mxArray.
-	mxArray* to_mex_handle() {
-		mxArray* handle = mxCreateNumericMatrix(1, 1, mxUINT32OR64_CLASS, mxREAL);
-		*reinterpret_cast<ObjectHandle<T>**> (mxGetPr(handle)) = this;
-		return handle;
-	}
-
-	string type_name() const {
-		return type->name();
-	}
-
-	// Convert mxArray (passed to mex-function) to an ObjectHandle<T>.
-	// Import a handle from MatLab as a mxArray of UINT32. Check that
-	// it is actually a pointer to an ObjectHandle<T>.
-	static ObjectHandle* from_mex_handle(const mxArray* handle) {
-		if (mxGetClassID(handle) != mxUINT32OR64_CLASS || mxIsComplex(handle)
-				|| mxGetM(handle) != 1 || mxGetN(handle) != 1) error(
-				"Parameter is not an ObjectHandle type.");
-
-		// We *assume* we can store ObjectHandle<T> pointer in the mxUINT32 of handle
-		ObjectHandle* obj = *reinterpret_cast<ObjectHandle**> (mxGetPr(handle));
-
-		if (!obj) // gross check to see we don't have an invalid pointer
-		error("Parameter is NULL. It does not represent an ObjectHandle object.");
-		// TODO: change this for max-min check for pointer values
-
-		if (obj->signature != obj) // check memory has correct signature
-		error("Parameter does not represent an ObjectHandle object.");
-
-		/*
-		 if (*(obj->type) != typeid(T)) { // check type
-		 mexPrintf("Given: <%s>, Required: <%s>.\n", obj->type_name(), typeid(T).name());
-		 error("Given ObjectHandle does not represent the correct type.");
-		 }
-		 */
-
-		return obj;
-	}
-
-	friend class Collector<T> ; // allow Collector access to signature
-};
-
-// --------------------------------------------------------- 
-// ------------------ Garbage Collection -------------------
-// --------------------------------------------------------- 
-
-// Garbage collection singleton (one collector object for each type T).
-// Ensures that registered handles are deleted when the dll is released (they
-// may also be deleted previously without problem).
-//    The Collector provides protection against resource leaks in the case
-// where 'clear all' is called in MatLab. (This is because MatLab will call
-// the destructors of statically allocated objects but not free-store allocated
-// objects.)
-template <typename T>
-class Collector {
-  typedef ObjectHandle<T> Handle;
-  typedef std::list< Handle* > ObjList;
-  typedef typename ObjList::iterator iterator;
-  ObjList objlist;
-public:
-  ~Collector() {
-    for (iterator i= objlist.begin(); i!=objlist.end(); ++i) {
-      if ((*i)->signature == *i) // check for valid signature
-	delete *i;
-    }
-  }
-
-  static void register_handle (Handle* obj) {
-    static Collector singleton;
-    singleton.objlist.push_back(obj);
-  }
-
-private: // prevent construction
-  Collector() {}
-  Collector(const Collector&);
-};
-
-//*****************************************************************************
-// wrapping C++ objects in a MATLAB proxy class
-//*****************************************************************************
-
-/* 
- For every C++ class Class, a matlab proxy class @Class/Class.m object
- is created. Its constructor will check which of the C++ constructors
- needs to be called, based on nr of arguments. It then calls the
- corresponding mex function new_Class_signature, which will create a
- C++ object using new, and pass the pointer to wrap_constructed
- (below). This creates a mexhandle and returns it to the proxy class
- constructor, which assigns it to self. Matlab owns this handle now.
-*/
-template <typename Class>
-mxArray* wrap_constructed(Class* pointer, const char *classname) {
-  ObjectHandle<Class>* handle = new ObjectHandle<Class>(pointer);
-  return handle->to_mex_handle();
-}
-
-/*
- [create_object] creates a MATLAB proxy class object with a mexhandle
- in the self property. Matlab does not allow the creation of matlab
- objects from within mex files, hence we resort to an ugly trick: we
- invoke the proxy class constructor by calling MATLAB, and pass 13
- dummy arguments to let the constructor know we want an object without
- the self property initialized. We then assign the mexhandle to self.
-*/
-// TODO: think about memory
-mxArray* create_object(const char *classname, mxArray* h) {
-  mxArray *result;
-  mxArray* dummy[13] = {h,h,h,h,h, h,h,h,h,h, h,h,h};
-  mexCallMATLAB(1,&result,13,dummy,classname);
-  mxSetProperty(result, 0, "self", h);
-  return result;
-}
-
-/*
- When the user calls a method that returns a shared pointer, we create
- an ObjectHandle from the shared_pointer and return it as a proxy
- class to matlab.
-*/
-template <typename Class>
-mxArray* wrap_shared_ptr(shared_ptr< Class > shared_ptr, const char *classname) {
-  ObjectHandle<Class>* handle = new ObjectHandle<Class>(shared_ptr);
-  return create_object(classname,handle->to_mex_handle());
-}
-
-//*****************************************************************************
-// unwrapping a MATLAB proxy class to a C++ object reference
-//*****************************************************************************
-
-/*
- Besides the basis types, the only other argument type allowed is a
- shared pointer to a C++ object. In this case, matlab needs to pass a
- proxy class object to the mex function. [unwrap_shared_ptr] extracts
- the ObjectHandle from the self property, and returns a shared pointer
- to the object.
-*/
-template <typename Class>
-shared_ptr<Class> unwrap_shared_ptr(const mxArray* obj, const string& className) {
-  bool isClass = mxIsClass(obj, className.c_str());
-  if (!isClass) {
-    mexPrintf("Expected %s, got %s\n", className.c_str(), mxGetClassName(obj));
-    error("Argument has wrong type.");
-  }
-  mxArray* mxh = mxGetProperty(obj,0,"self");
-  if (mxh==NULL) error("unwrap_reference: invalid wrap object");
-  ObjectHandle<Class>* handle = ObjectHandle<Class>::from_mex_handle(mxh);
-  return handle->get_object();
-}
-
-//*****************************************************************************
diff --git a/wrap/wrap.cpp b/wrap/wrap.cpp
deleted file mode 100644
index 1a52da6bd..000000000
--- a/wrap/wrap.cpp
+++ /dev/null
@@ -1,58 +0,0 @@
-/* ----------------------------------------------------------------------------
-
- * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
- * Atlanta, Georgia 30332-0415
- * All Rights Reserved
- * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
-
- * See LICENSE for the license information
-
- * -------------------------------------------------------------------------- */
-
-/**
- * file: wrap.ccp
- * brief: wraps functions
- * Author: Frank Dellaert
- **/
-
-#include <stdio.h>
-#include <iostream>
-
-#include "Module.h"
-
-using namespace std;
-
-/* ************************************************************************* */
-/**
- * main function to wrap a module
- */
-void generate_matlab_toolbox(const string& interfacePath,
-			     const string& moduleName,
-			     const string& toolboxPath,
-			     const string& nameSpace,
-			     const string& mexFlags) 
-{
-  // Parse into class object
-  Module module(interfacePath, moduleName, false);
-
-  // emit MATLAB code
-  module.matlab_code(toolboxPath,nameSpace,mexFlags);
-}
-
-/* ************************************************************************* */
-
-int main(int argc, const char* argv[]) {
-  if (argc<5 || argc>6) {
-    cerr << "wrap parses an interface file and produces a MATLAB toolbox" << endl;
-    cerr << "usage: wrap interfacePath moduleName toolboxPath" << endl;
-    cerr << "  interfacePath : *absolute* path to directory of module interface file" << endl;
-    cerr << "  moduleName    : the name of the module, interface file must be called moduleName.h" << endl;
-    cerr << "  toolboxPath   : the directory in which to generate the wrappers" << endl;
-    cerr << "  nameSpace     : namespace to use, pass empty string if none" << endl;
-    cerr << "  [mexFlags]    : extra flags for the mex command" << endl;
-  }
-  else
-    generate_matlab_toolbox(argv[1],argv[2],argv[3],argv[4],argc==5 ? " " : argv[5]);
-}
-
-/* ************************************************************************* */