Quaternion

Definition

$$\mathbf{q}=q_w+q_x\mathbf{i}+q_y\mathbf{j}+q_z\mathbf{k},{\mathbf{i}}^2={\mathbf{j}}^2={\mathbf{k}}^2=\mathbf{ijk}=-1$$

Unit quaternions ($\Vert \mathbf{q}\Vert =1$, the 3-sphere $S^3$) represent rotations. As scalar + vector, $\mathbf{q}=[q_w\mathbf{v}{]}^T$.

Formula

From axis $\omega$ and angle $\theta$:

$$q_w=\cos \frac{\theta }{2},\mathbf{v}=\sin \frac{\theta }{2}\omega$$

Rotate a vector: ${\mathbf{a}}^{\prime }=\mathbf{q}\circ \bar{\mathbf{a}}\circ \bar{\mathbf{q}}$ with the conjugate $\bar{\mathbf{q}}=(q_w,-\mathbf{v})$. Interpolate: $\text{Slerp}(q_0,q_1,t)=q_0(q_0^{-1}{q}_1{)}^t$, $t\in [0,1]$.

Intuition

Complex numbers multiply by adding angles and multiplying magnitudes, so a unit complex number is a 2D rotation. Quaternions are Hamilton’s generalization of this to 3D rotations, which needs 4 components.

Pro	Con
no singularities (no gimbal lock)	4 numbers instead of 3
linearly independent derivatives	quadratic constraint $\Vert \mathbf{q}\Vert =1$
concatenation with the quaternion product
ideal for interpolation (SLERP)

Appears in

• Lecture 2.2, Quaternions: complex numbers, quaternion product, SLERP, pros and cons
• Lecture 2.2, Which representation: best for singularities and concatenation
• Lecture 8, 3DGS covariance: $\Sigma =RS{S}^T{R}^T$ with $R$ from a quaternion

Related concepts

• Gaussian Splatting
• Rotation Matrix
• Euler Angles
• Axis-Angle and Exponential Map