Quaternions

• Rotation matrix: Describes 3 DOF with 9 quantities
• Euler angles and the rotation vectors: Compact but suffer from the singularity

Quaternions are a way to represent rotations. Conceptually they are similar to complex numbers.

Complex Numbers for 2D Rotation

We use the complex set $\mathbb{C}$ to represent the vector on a 2D complex plane, and the complex multiplication with a unit complex number represents rotation on the 2D plane.

• Multiplying by the complex $i$ is equivalent to rotating a complex vector counteclockwise by $90°$.

For example, when we want to rotate the vector of a complex plane by $\theta$, we can multiply this complex vector by $e^{i\theta }$, which is represented by polar coordinates. It can also be written in the form of the Euler equation:

$$e^{i\theta }=\cos \theta +i\sin \theta$$

This is a unit-length complex number. Therefore, in two dimensions, the rotation can be described by a unit complex number.

Definition

Quaternions are basically a 3D version of the idea above. A quaternion $\mathbf{q}$ has a real part and three imaginary parts:

$$\mathbf{q}=q_0+q_{1}i+q_{2}j+q_{3}k$$ $$\{\begin{array}{l}{i}^2=j^2=k^2=1\\ ij=k,ji=-k\\ jk=i,kj=-1\\ ki=j,ik=-j\end{array}$$

If we look at $i,j,k$ as three axes, they look the same as complex numbers when multiplying with themselves, and the same as the outer product when multiplying with the others.

We can also use a scalar and a vector to express quaternions:

$$\mathbf{q}=[s,\mathbf{v}{]}^T,s=q_0\in \mathbb{R},\mathbf{v}=[q_1,q_2,q_3{]}^T\in {\mathbb{R}}^3$$

Here, $s$ is the real part of the quaternion, and $\mathbf{v}$ is its imaginary part. If the imaginary part of a quaternion is 0, it is called real quaternion. Conversely, if its real part is 0, it is called imaginary quaternion.

We can use a unit quaternion to represent any rotation in 3D space, but it’s not the same as complex numbers. In the complex, multiplying by $i$ means rotating by $90°$. In quaternion form, multiplied by $i$ is rotating around the $i$ axis by $90°$? So, does $ij=k$ means, first rotating around the $i$ by $90°$, then around $j$ by $90°$, is equivalent to rotating around $k$ by $90°$? This is not the case. the correct case would be that multiplying $i$ corresponds to rotating $180°$, to guarantee that $ij=k$. Thus, $i^2=-1$ means that after rotating $360°$ around the $i$ axis, we get the opposite thing. This object has to be rotated by $720°$ to be equal to its original appearance.

Using Quaternions to Represent a Rotation

We can use a quaternion to express the rotation of a point. Suppose a point in 3D space $\mathbf{p}=[x,y,z{]}^T\in \mathbb{R}$ and rotation is specified by a unit quaternion $\mathbf{q}$. The 3D point is rotated to become ${\mathbf{p}}^{\prime }$. If we use a Rotation Matrix, then there is ${\mathbf{p}}^{\prime }=\mathbf{Rp}$. And if we use quaternion to describe rotation, how do we operate a 3D vector with a quaternion?

First, we extend the 3D point to an imaginary quaternion:

$$\mathbf{p}=[0,x,y,z{]}^T=[0,\mathbf{v}{]}^T$$

We just put the 3 coordinates into the imaginary part and leave the real part to be zero. Then, the rotated point ${\mathbf{p}}^{\prime }$ can be expressed as such a a product:

$${\mathbf{p}}^{\prime }={\mathbf{qpq}}^{-1}$$

The multiplication here is the quaternion multiplication, and the result is also a quaternion. Finally, we take the imaginary part of ${\mathbf{p}}^{\prime }$ and get the coordinates of the point after the rotation. It can be easily verified (we leave as an exercise here) that the real part of the calculation is $0$, so it is a pure imaginary quaternion.