Intro to Lie groups for rigid transformations

1 Introduction

A Lie group is a group that deals with manifolds. What is a manifold and what is a group?

When we think about geometry, most of us think about geometry in a Euclidean space. A Euclidean space has many nice properties. One property is that, if you travel in one direction, you keep going in that direction, and never find yourself going in another direction. A good example is the 2D Euclidean space \mathbb R^2. If you’re travelling in the x-direction, the y-coordinate couldn’t care less!

In a Euclidean space, you can also keep travelling in a particular direction and never come back to where you started.

However, there are some spaces that are non-Euclidean. For example, imagine standing on the surface of a sphere. If you’re standing in San Francisco facing north, and you keep walking forwards, you’ll eventually find yourself in Kazakhstan, which is in the Eastern hemisphere, even though you started out in the West! And then if you go even further, you’ll end up right where you started!

cover image of The Little Prince
FIGURE 1 The Little Prince stands on a spherical surface which is a manifold. He can walk forwards, backwards, left, and right as though he’s on a flat plane, but in the end the sphere is not a plane.

A manifold is a space that may not be Euclidean, but sure looks Euclidean in a small area around you. In the sphere example, the small area around you looks pretty much just like a flat plane, which is Euclidean! Cue the flat Earther jokes!

A group is a set that has some operation \oplus to combine two things in the group to make a third thing, so that:

A matrix group uses matrix multiplication for \oplus so that it automatically satisfies the last three requirements.

A Lie group is a group that is also associated with a Lie algebra to help us deal with manifolds. If we proceed with the analogy of standing on a the spherical Earth…

Before we get into details, let us begin with an example.

2 Rotations in 2D

Now we can start with an example of a classical Lie group called SO(2), which stands for the special orthogonal group. Put simply, this is a 2D rotation.

2D rotations are not Euclidean, because if you rotate by 360 degrees, you’ll end up at the same place, which is something that cannot happen in a Euclidean space. However, locally, they are like \mathbb R^1, a trivial 1-dimensional Euclidean space.

Imagine a point

1 \displaystyle{\begin{align}\mathbf p = \begin{bmatrix}x\\y\end{bmatrix}\end{align}}

and we want to rotate it anticlockwise a little bit, by a small angle \omega. How does the point’s position change?

Well, first we study how it changes depending on the point’s x position.

how a point on the x-axis moves
FIGURE 2 How a point’s position changes based on its x-position.

As you can see, the more the x-coordinate, the more the y-coordinate increases. Yet, the way its x-coordinate changes doesn’t depend on its previous x-coordinate at all.

So we can write:

2 \displaystyle{\begin{align}y_\text{new} = y + \omega x\end{align}}

Likewise, the more the y-coordinate, the more the x-coordinate decreases.

how a point moves when rotated in 2D
FIGURE 3 How a point’s position changes when rotated in 2D.

So we can write:

3 \displaystyle{\begin{align}x_\text{new} = x - \omega y\end{align}}

In matrix notation,

4 \displaystyle{\begin{align}\begin{bmatrix}x_\text{new}\\y_\text{new}\end{bmatrix} = \begin{bmatrix}x - \omega y\\y + \omega x\end{bmatrix} = \underbrace{\left(\mathbf I + \omega \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix} \right)}_A \mathbf p\end{align}}

Well, that’s cool, but we’ve only rotated it by a tiny amount. To rotate it by a nontrivial amount, we need to do it a bunch of times!

moving a point several times
FIGURE 4 Moving a point several times by applying the same transformation repeatedly.

Suppose we actually want to rotate things by an angle \theta. We can make the tiny amount \omega be a small fraction of \theta, say,

5 \displaystyle{\begin{align}\omega = \frac{\theta}{n}\end{align}}

Then, to rotate it by the full angle \theta, we’d need to apply it n times. The biggger the n, the better!

6 \displaystyle{\begin{align}\mathbf R \mathbf p = \lim_{n\rightarrow \infty} \left(\mathbf I + \frac{\theta}{n} \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\right)^n \mathbf p\end{align}}

Let’s define the generator \mathbf G:

7 \displaystyle{\begin{align}\mathbf G = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\end{align}}

So the rotation matrix is:

8 \displaystyle{\begin{align}\mathbf R = \lim_{n\rightarrow \infty} \left(\mathbf I + \frac{\theta \mathbf G}{n}\right)^n\end{align}}

To find the limit, we can use the binomial theorem:

9 \begin{align}\left(\mathbf I + \frac{\theta \mathbf G}{n}\right)^n = \mathbf I
&+ \frac{n}{1!}\left(\frac{\theta \mathbf G}{n}\right)^1\\
&+ \frac{n(n-1)}{2!}\left(\frac{\theta \mathbf G}{n}\right)^2\\
&+ \frac{n(n-1)(n-2)}{3!}\left(\frac{\theta \mathbf G}{n}\right)^3\\
&+ \cdots\end{align}

But then as n approaches infinity, all the ns cancel out. We use the monotone convergence theorem and apply the limit to each term, e.g. the third one:

10 \displaystyle{\begin{align}\lim_{n\rightarrow \infty} \frac{n(n-1)(n-2)}{n^3} = 1\end{align}}

Now we’re left with:

11 \displaystyle{\begin{align}\mathbf R = \mathbf I + \theta \mathbf G + \frac{(\theta \mathbf G)^2}{2!} + \frac{(\theta \mathbf G)^3}{3!} + \cdots\end{align}}

Notice it is the Taylor expansion of the exponential function.

This is called the exponential map, and the Taylor series definition applies to all Lie groups.

12 \displaystyle{\begin{align}\mathbf R \equiv \exp{(\theta \mathbf G)}\end{align}}

In the case of 2D rotations, we can sum up this series with a closed form.

Notice that:

13 \displaystyle{\begin{align}\mathbf G^2 = -\mathbf I.\end{align}}

Then we group odd and even terms and notice they resemble the Taylor series of cosine and sine,

14 \begin{align}
\mathbf R &= \mathbf I + \theta \mathbf G + \frac{(\theta \mathbf G)^2}{2!} + \frac{(\theta \mathbf G)^3}{3!} + \cdots\\
&= \mathbf I + \theta\mathbf G - \frac{\theta^2 \mathbf I}{2!} - \frac{\theta^3 \mathbf G}{3!} + \frac{\theta^4\mathbf I}{4!} + \frac{\theta^5\mathbf G}{5!} \cdots\\
&= \left(1 + \frac{\theta^2}{2!} + \frac{\theta^4}{4!} \cdots\right)\mathbf I + \left(\frac{\theta}{1!} - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} -\cdots\right) \mathbf G\\
&= \mathbf I \cos\theta + \mathbf G \sin \theta\\
&= \begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin\theta & \cos\theta

Hooray, we’ve just derived the rotation matrix!

To recap:

The astute reader will notice many similarities between this exercise and imaginary numbers. Indeed, an imaginary number \theta i, representing an angle, is related to rotation in the form of \exp(\theta i) in much the same way.

3 Rotations in 3D

Unlike in 2D, we can now rotate things in three axes.

coordinate system
FIGURE 5 We can rotate things in three axes in 3D.

Based on the same idea as previously, we can think about what happens if we rotate stuff slightly by \theta around the z-axis. Well, the point’s z-coordinate remains unchanged, but its x- and y-coordinates change in exactly the same way as previously.

15 \displaystyle{\begin{align}\mathbf G_z = \begin{bmatrix}0 & -1 & 0\\
1 & 0 & 0 \\
0 & 0& 0

Notice the top-left 2\times 2 submatrix is exactly the same as in the previous section.

We can likewise derive:

16 \displaystyle{\begin{align}\mathbf G_x = \begin{bmatrix}0 & 0 & 0\\
0 & 0 & -1\\
0 & 1 & 0\end{bmatrix},
\mathbf G_y = \begin{bmatrix}0 & 0 & 1\\
0 & 0 & 0\\
-1 & 0 & 0

Then, an element \boldsymbol \omega_\times \in \mathfrak{so}(3) is the skew-symmetric matrix:

17 \begin{align} \boldsymbol \omega_\times &= \omega_x \mathbf G_x + \omega_y \mathbf G_y + \omega_z \mathbf G_z \\
&= \begin{bmatrix}0 & -\omega_3 & \omega_2\\
\omega_3 & 0 & -\omega_1\\
-\omega_2 & \omega_1 & 0\end{bmatrix}.

Note that the matrix \boldsymbol \omega_\times is called the “cross-product matrix” because

18 \displaystyle{\begin{align}\boldsymbol \omega_\times \mathbf p =
\begin{bmatrix}\omega_1\\\omega_2\\\omega_3\end{bmatrix}\times \mathbf p = \boldsymbol \omega \times \mathbf p.\end{align}}

The exponential can be computed again by grouping odd and even terms of the Taylor series.

19 \begin{align}
\mathbf R = \exp(\boldsymbol \omega_\times)
&\equiv \mathbf I + \boldsymbol \omega_\times + \frac{\boldsymbol\omega_\times^2}{2!} + \frac{\boldsymbol\omega_\times^3}{3!} + \cdots\\
&=\mathbf I + \boldsymbol \omega_\times \frac{\sin \|\boldsymbol \omega\|}{\|\boldsymbol \omega\|}
+ \boldsymbol \omega_\times^2 \frac{1 - \cos \|\boldsymbol \omega\|}{\|\boldsymbol \omega\|^2}.

This is called the Rodrigues’ rotation formula.

The vector \boldsymbol \omega is also called the axis-angle representation because its norm \|\boldsymbol \omega\| represents the angle, while the normalised vector is the axis of rotation.

Interestingly, in all the examples we’ve looked at so far, we’re dealing with spheres whose surfaces look like planes of one fewer dimension.

Just like how a 3D sphere has x^2 + y^2 + z^2 = 1, a unit quaternion representing must have q_x^2 + q_y^2 + q_z^2 + q_w^2 = 1.

3.1 Implementations of 3D rotations

The Lie Group SO(3) is a group of rotation matrices, but we can implement it in many ways.

For example we could imagine there being an abstract trait SO3 in your favourite programming language, supporting common operators such as rotate_point, compose, log, and so on.

The following is Rust-like pseudocode.

trait SO3 {
    fn from_so3(so3: &Point3) -> Self;
    fn rotate_point(&self, point: &Point3) -> Point3;
    fn compose(&self, other: &Self) -> Self;
    /* ... and so on ...*/

impl SO3 for Mat3 {
    fn from_so3(so3: &Point3) -> Self {
        // TODO: implement Rodrigues formula here lol
    fn rotate_point(&self, point: &Point3) -> Point3 {
        // using matrix multiplication
        self * point
    fn compose(&self, other: &Self) -> Self {
        // using matrix multiplication
        self * other

impl SO3 for Quaternion {
    fn from_so3(so3: &Point3) -> Self {
        let half_theta = so3.norm() / 2.0;
        let s = theta.sin();
        let c = theta.cos();
        Self::new(s * so3.x, s * so3.y, s * so3.z, c)
    fn rotate_point(&self, point: &Point3) -> Point3 {
        // using quaternion multiplication
        let q = Quaternion::new(point.x, point.y, point.z, 0.0);
        self.conjugate() * q * self
    fn compose(&self, other: &Self) -> Self {
        // using quaternion multiplication
        self * other

There are many other ways to implement SO3 as explained on Wikipedia. Common ways to implement SO3 could include:

Which parameterisation of SO(3) to use depends on what you want to do.

4 Rigid transformations SE(3)

The Special Euclidean Group SE(3) deals with rigid transformations by combining a rotation in SO(3) with a translation in \mathbb R^3.

20 \displaystyle{\begin{align}\mathbf T_{4\times 4} = \begin{bmatrix}
\mathbf R_{3\times 3} & \mathbf t_{3\times 1}\\
\mathbf 0_{1\times 3} & 1
\end{bmatrix} \in SE(3).\end{align}}

Just like SO(2) and SO(3), each element in SE(3) is associated with an element on the corresponding Lie algebra, \mathfrak{se}(3):

21 \begin{align}
\boldsymbol\xi_{4\times 4} &= \begin{bmatrix}
[\boldsymbol\omega]_\times & \boldsymbol\tau\\
\mathbf 0_{1 \times 3} & 0
\end{bmatrix} \in \mathfrak{se}(3)\\
&= \begin{bmatrix}
0 & -\omega_3 & \omega_2 & \tau_1\\
\omega_3 & 0 & -\omega_1 & \tau_2\\
-\omega_2 & \omega_1 & 0 & \tau_3\\
0 & 0 & 0 & 0
\end{bmatrix} \in \mathfrak{se}(3)

where [\boldsymbol\omega]_\times is the cross product matrix of \omega. We may also write it as a 6\times 1 vector.

22 \displaystyle{\begin{align}\boldsymbol\xi_{6\times 1} = \begin{bmatrix}
\boldsymbol\omega_{3\times 1}\\
\boldsymbol\tau_{3\times 1}

Some textbooks put the rotational part on top and some put it on the bottom; it doesn’t matter as long as you’re consistent.

In future blog posts we will use the 6\times 1 and 4\times 4 representations interchangeably depending on context.

A transformation \mathbf T \in SE(3) is used to transform a 3D point \mathbf p \in \mathbb R^3:

23 \displaystyle{\begin{align}\mathbf p_{3\times 1} = \begin{bmatrix}x\\y\\z\end{bmatrix}.\end{align}}

The homogeneous representation adds a one at the end:

24 \displaystyle{\begin{align}\mathbf p_{4\times 1} = \begin{bmatrix}x\\ y\\ z\\ 1\end{bmatrix}\end{align}}

so that points may be transformed rigidly

25 \displaystyle{\begin{align}\mathbf T \mathbf p \equiv \mathbf R \mathbf p + \mathbf t.\end{align}}

5 Things to do with Lie groups

Lie groups provide us with elegant tools to do:

In a next blog post we will cover optimization on the manifold of Lie groups for all kinds of robotics applications like SLAM!