Lie Group fro 2D transformations [1]

$\mathrm{SO}(2)$ : Rotations in 2D space

Having treated $\mathrm{SO}(3)$, the 2D equivalent $\mathrm{SO}(3)$ is straightforward.

Representation

Elements of the rotation group in 2D, $\mathrm{SO}(2)$, are represented by 2D rotation matrices. Composition and inversion in the group correspond to matrix multiplication and inversion. Because rotation matrices are orthogonal, inversion is equivalent to transposition.

$$\begin{array}{rl}\mathbf{R}& \in \mathrm{SO}(2)\\ {\mathbf{R}}^{-1}& ={\mathbf{R}}^{\mathrm{\top }}\end{array}$$

The Lie algebra, $\mathfrak{so}(2)$, is the set of $2\times 2$ skew-symmetric matrices. The single generator of $\mathfrak{so}(2)$ corresponds to the derivative of 2D rotation, evaluated at the identity:

$$\mathbf{J}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

An element of $\mathfrak{so}(2)$ is then any scalar multiple of the generator:

$$\begin{array}{rl}\theta & \in \mathbb{R}\\ \theta \mathbf{J}& \in \mathfrak{so}(2)\end{array}$$

We will simply write $\theta \in \mathfrak{so}(2)$, and use ${\theta }_{\times }$to represent the skew symmetric matrix $\theta \mathbf{J}$.

Exponential Map

The exponential map that takes skew symmetric matrices to rotation matrices is simply the matrix exponential over a linear combination of the generators:

$$\begin{array}{rl}\exp \left({\theta }_{\times }\right)& \equiv \exp \left(\begin{array}{cc}0& -\theta \\ \theta & 0\end{array}\right)\\ & =\mathbf{I}+{\theta }_{\times }+\frac{1}{2!}{\theta }_{\times }^2+\frac{1}{3!}{\theta }_{\times }^3+\cdots \\ & =\mathbf{I}+\left(\begin{array}{cc}0& -\theta \\ \theta & 0\end{array}\right)+\frac{1}{2!}\left(\begin{array}{cc}-{\theta }^2& 0\\ 0& -{\theta }^2\end{array}\right)+\frac{1}{3!}\left(\begin{array}{cc}0& {\theta }^3\\ -{\theta }^3& 0\end{array}\right)\end{array}$$

The resulting elements form the Taylor series expansion of $\sin \theta$ and $\cos \theta$ :

$$\begin{array}{rl}\exp \left({\theta }_{\times }\right)& =\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\in \mathrm{SO}(2)\\ & =\cos \theta \mathbf{I}+\sin \theta \mathbf{J}\end{array}$$

Thus the exponential map yields a rotation by $\theta$ radians. The exponential map can be inverted, going from $\mathrm{SO}(2)$ to $\mathfrak{so}(2)$ :

$$\begin{array}{rl}\mathbf{R}& \in \mathrm{SO}(2)\\ \text{(1)}& \ln (\mathbf{R})=\theta & =\arctan ({\mathbf{R}}_{21},{\mathbf{R}}_{11})\end{array}$$

Adjoint

Because rotations in the plane commute, the adjoint of $\mathrm{SO}(2)$ is the identity function.

SE(2): Rigid transformations in 2D space

The group $\mathrm{SE}(2)$ is the lower-dimensional analogue of $\mathrm{SE}(3)$. The group has 3D, corresponding to translation and rotation in the plane.

Representation

The group of rigid transformations in 2D space, $\mathrm{SE}(2)$, is represented by linear transformations on homogeneous three-vectors:

$$\begin{array}{rl}\mathbf{R}& \in \mathrm{SO}(2),\mathbf{t}\in {\mathbb{R}}^2\\ C& =\left(\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ \mathbf{0}& 1\end{array}\right)\in \mathrm{SE}(2)\end{array}$$

Note that, in an implementation, only $\mathbf{R}$ and $\mathbf{t}$ need to be stored. The remaining matrix structure can remain implicit.

Transformation composition and inversion are coincident with matrix multiplication and inversion:

$$\begin{array}{rl}{C}_1,C_2& \in \mathrm{SE}(2)\\ C_1\cdot C_2& =\left(\begin{array}{cc}{\mathbf{R}}_1& {\mathbf{t}}_1\\ \mathbf{0}& 1\end{array}\right)\cdot \left(\begin{array}{cc}{\mathbf{R}}_2& {\mathbf{t}}_2\\ \mathbf{0}& 1\end{array}\right)\\ & =\left(\begin{array}{cc}{\mathbf{R}}_1{\mathbf{R}}_2& {\mathbf{R}}_1{\mathbf{t}}_2+{\mathbf{t}}_1\\ \mathbf{0}& 1\end{array}\right)\\ C_1^{-1}& =\left(\begin{array}{cc}{\mathbf{R}}_1^{\mathrm{\top }}& -{\mathbf{R}}_1^{\mathrm{\top }}\mathbf{t}\\ \mathbf{0}& 1\end{array}\right)\end{array}$$

The matrix representation also makes the group action on 2D points and vectors explicit:

$$\begin{array}{rl}\mathbf{x}& ={\left(\begin{array}{lll}x& y& w\end{array}\right)}^{\mathrm{\top }}\in {\mathbb{R}\mathbb{P}}^2(\lambda \mathbf{x}\simeq \mathbf{x}\mathrm{\forall }\lambda \in \mathbb{R})\\ C\cdot \mathbf{x}& =\left(\begin{array}{ll}\mathbf{R}& \mathbf{t}\\ \mathbf{0}& 1\end{array}\right)\cdot \mathbf{x}\\ & =\left(\begin{array}{c}\mathbf{R}{\left(\begin{array}{ll}x& y\end{array}\right)}^{\mathrm{\top }}+w\mathbf{t}\\ w\end{array}\right)\end{array}$$

Typically, $w=1$, so that $\mathbf{x}$ is a Cartesian point. The action by matrix-vector multiplication corresponds to first rotating $\mathbf{x}$ and then translating it. For direction vectors, encoded with $w=0$, translation is ignored.

The Lie algebra $\mathfrak{se}(2)$ is the set of $3\times 3$ matrices corresponding to differential translations and rotation around the identity. There are thus three generators of the algebra:

$$G_1=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),G_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),G_3=\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right)$$

An element of $\mathfrak{se}(2)$ is then represented by linear combinations of the generators:

$$\begin{array}{rl}{\left(\begin{array}{cc}{\mathbf{u}}_1& {\mathbf{u}}_2\theta \end{array}\right)}^{\mathrm{\top }}& \in {\mathbb{R}}^3\\ {\mathbf{u}}_1{G}_1+{\mathbf{u}}_2{G}_2+\theta G_3& \in \mathfrak{se}(2)\end{array}$$

For convenience, we write $(\mathbf{u}\theta {)}^{\mathrm{\top }}\in \mathfrak{se}(2)$, with multiplication against the generators implied.

Exponential Map

As for all Lie groups in this document, the exponential map from $\mathfrak{se}(2)$ to $\mathrm{SE}(2)$ is the matrix exponential on a linear combination of the generators:

$$\begin{array}{rl}\mathit{\delta }=\left(\begin{array}{ll}\mathbf{u}& \theta \end{array}\right)& \in \mathfrak{se}(2)\\ \exp (\mathit{\delta })& =\exp \left(\begin{array}{cc}{\theta }_{\times }& \mathbf{u}\\ \mathbf{0}& 0\end{array}\right)\\ & =\mathbf{I}+\left(\begin{array}{cc}{\theta }_{\times }& \mathbf{u}\\ \mathbf{0}& 0\end{array}\right)+\frac{1}{2!}\left(\begin{array}{cc}{\theta }_{\times }^2& {\theta }_{\times }\mathbf{u}\\ \mathbf{0}& 0\end{array}\right)+\frac{1}{3!}\left(\begin{array}{cc}{\theta }_{\times }^3& {\theta }_{\times }^2\mathbf{u}\\ \mathbf{0}& 0\end{array}\right)+\cdots \end{array}$$

The rotation block is the same as for $\mathrm{SO}(2)$, but the translation component is a different power series:

$$\begin{array}{rl}\exp \left(\begin{array}{cc}{\theta }_{\times }& \mathbf{u}\\ \mathbf{0}& 0\end{array}\right)& =\left(\begin{array}{cc}\exp \left({\theta }_{\times }\right)& \mathbf{V}\mathbf{u}\\ \mathbf{0}& 1\end{array}\right)\\ \mathbf{V}& =\mathbf{I}+\frac{1}{2!}{\theta }_{\times }+\frac{1}{3!}{\theta }_{\times }^2+\cdots \end{array}$$

We split the terms by odd and even powers:

$$\mathbf{V}=\sum _{i=0}^{\mathrm{\infty }}[\frac{{\theta }_{\times }^{2i}}{(2i+1)!}+\frac{{\theta }_{\times }^{2i+1}}{(2i+2)!}]$$

Two identities (easily confirmed by induction) are useful for collapsing the series:

$$\begin{array}{rl}{\theta }_{\times }^{2i}& =(-1{)}^i{\theta }^{2i}\cdot \left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)\\ {\theta }_{\times }^{2i+1}& =(-1{)}^i{\theta }^{2i+1}\cdot \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\end{array}$$

Direct application of the identies yields a reduced expression for $\mathbf{V}$ in terms of diagonal and skewsymmetric components:

$$\begin{array}{rl}\mathbf{V}& =\sum _{i=0}^{\mathrm{\infty }}(-1{)}^i{\theta }^{2i}[\frac{1}{(2i+1)!}\cdot \left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)+\frac{\theta }{(2i+2)!}\cdot \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)]\\ & =\left(\sum _{i=0}^{\mathrm{\infty }}\frac{(-1{)}^i{\theta }^{2i}}{(2i+1)!}\right)\cdot \left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)+\left(\sum _{i=0}^{\mathrm{\infty }}\frac{(-1{)}^i{\theta }^{2i+1}}{(2i+2)!}\right)\cdot \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\end{array}$$

The coefficients can be identified with Taylor expansions:

$$\begin{array}{rl}\mathbf{V}& =(1-\frac{{\theta }^2}{3!}+\frac{{\theta }^4}{5!}+\cdots )\cdot \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+(\frac{\theta }{2!}-\frac{{\theta }^3}{4!}+\frac{{\theta }^5}{6!}+\cdots )\cdot \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\\ & =\left(\frac{\sin \theta }{\theta }\right)\cdot \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\frac{1-\cos \theta }{\theta }\right)\cdot \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\\ & =\frac{1}{\theta }\cdot \left(\begin{array}{cc}\sin \theta & -(1-\cos \theta )\\ 1-\cos \theta & \sin \theta \end{array}\right)\end{array}$$

For implementation purposes, Taylor expansions should be used for $\mathbf{V}$ when $\theta$ is small. The $\ln ()$ function on $\mathrm{SE}(2)$ can be implemented by first recovering $\theta =\ln (\mathbf{R})$ as shown in Eq. (1), then solving $\mathbf{V}\mathbf{u}=\mathbf{t}$ for $\mathbf{u}$ in closed form:

$$\begin{array}{rl}A& \equiv \frac{\sin \theta }{\theta }\\ B& \equiv \frac{1-\cos \theta }{\theta }\\ {\mathbf{V}}^{-1}& =\frac{1}{A^2+B^2}\left(\begin{array}{cc}A& B\\ -B& A\end{array}\right)\\ \ln \left(\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ \mathbf{0}& 1\end{array}\right)& =(\genfrac{}{}{0ex}{}{{\mathbf{V}}^{-1}\cdot \mathbf{t}}{\theta })\in \mathfrak{se}(2)\end{array}$$

Adjoint

The adjoint in $\mathrm{SE}(2)$ is computed from the generators:

$$\begin{array}{rl}\mathit{\delta }={\left(\begin{array}{ll}\mathbf{u}& \theta \end{array}\right)}^{\mathrm{\top }}& \in \mathfrak{se}(2),C=\left(\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ \mathbf{0}& 1\end{array}\right)\in \mathrm{SE}(2)\\ {\mathrm{Adj}}_C\cdot \mathit{\delta }& =C\cdot \left(\sum _{i=1}^3{\mathit{\delta }}_i{G}_i\right)\cdot C^{-1}\\ & =(\genfrac{}{}{0ex}{}{\mathbf{R}\mathbf{u}+\theta (\genfrac{}{}{0ex}{}{{\mathbf{t}}_2}{-{\mathbf{t}}_1}))}{\theta })\\ ⟹{\mathrm{Adj}}_C& =\left(\begin{array}{cc}\mathbf{R}& {\mathbf{t}}_2\\ & -{\mathbf{t}}_1\\ \mathbf{0}& 1\end{array}\right)\in {\mathbb{R}}^{3\times 3}\end{array}$$

Note that moving a tangent vector via the adjoint mixes the rotation component into the translation component.

Ref

[1] Lie Groups for 2D and 3D Transformations, PDF, by Ethan Eade, 2017