Quaternion

A quaternion is an extension of complex numbers used to represent rotations and 3D orientation.

Definition §

A quaternion has the form

$$q=a+bi+cj+dk$$

where $a,b,c,d\in \mathbb{R}$ and the imaginary units satisfy

$$i^2=j^2=k^2=ijk=-1$$

The multiplication rules are

$$ij=k,jk=i,ki=j$$

and reversing the order changes the sign:

$$ji=-k,kj=-i,ik=-j$$

This means quaternion multiplication is not commutative.

Scalar and Vector Parts §

We can write

$$q=(a,\vec{v})$$

where the scalar part is $a$ and the vector part is

$$\vec{v}=(b,c,d)$$

So

$$q=a+\vec{v}$$

in shorthand.

Conjugate, Norm, and Inverse §

The conjugate of

$$q=a+bi+cj+dk$$

is

$$\bar{q}=a-bi-cj-dk$$

The norm is

$$|q|=\sqrt{a^2+b^2+c^2+d^2}$$

and

$$q\bar{q}=\bar{q}q=|q{|}^2$$

If $q\ne 0$, then the inverse is

$$q^{-1}=\frac{\bar{q}}{|q{|}^2}$$

Unit Quaternion §

A unit quaternion satisfies

$$|q|=1$$

Unit quaternions are especially important because they represent rotations in 3D.

Quaternion Multiplication §

If

$$q_1=(a,\vec{u}),q_2=(b,\vec{v})$$

then

$$q_1{q}_2=(ab-\vec{u}\cdot \vec{v},a\vec{v}+b\vec{u}+\vec{u}\times \vec{v})$$

This combines the dot product and cross product into one algebraic rule.

Rotations in 3D §

To rotate by angle $\theta$ around a unit axis $\hat{n}=(x,y,z)$, use the unit quaternion

$$q=\cos \left(\frac{\theta }{2}\right)+(xi+yj+zk)\sin \left(\frac{\theta }{2}\right)$$

A vector $\vec{p}=(p_x,p_y,p_z)$ is written as the pure quaternion

$$p=0+p_{x}i+p_{y}j+p_{z}k$$

The rotated vector is obtained by

$$p^{\prime }=qp{q}^{-1}$$

Why Quaternions Are Useful §

• They avoid gimbal lock that can happen with Euler angles
• They are more compact than rotation matrices
• They are numerically stable for interpolation
• They are widely used in computer graphics, robotics, and physics