Quaternions

Quaternions can be used to represent rotations.

• Require less memory
• Concatenating quaternions require less arithmetic
• More easily interpolated

They are represented as a 4D vector.

The 4th component is a scalar value, taking the space for $w$.

$$q = s + \vec{v} = \langle w, x, y, z \rangle$$

Alternative representation (Grimoire uses this notation). This notation makes it look like a vector in homogenous coordinates.

$$q = \langle x, y, z, w \rangle$$

Creating a quaternion from an angle and axis of rotation

$$q = cos \frac{\theta}{2} + \vec{A} sin \frac{\theta}{2}$$

where $\theta$ is an angle about the unit axis $A$.

Multiplying Quaternions

Multiplication is NOT commutative, so order really matters.

$$q_1 q_2 = s_1 s_2 - \vec{v_1} \cdot \vec{v_2} + s_1 \vec{v_2} + s_2 \vec{v_1} + \vec{v_1} \times \vec{v_2}$$

Rotating Vectors with Quaternions

Suppose some function $\phi$ represents a rotation meant to be applied to some point $\vec{P}$. To be a rotation it must:

• Preserve lengths
• Preserve angles
• Preserve handedness

Length is preserved if:

$$\lVert \phi(\vec{P}) \rVert = \lVert \vec{P} \rVert$$

Angle is preserved if:

$$\phi(\vec{P_1}) \cdot \phi(\vec{P_2}) = \vec{P_1} \cdot \vec{P_2}$$

Handed is preserved if:

$$\phi(\vec{P_1}) \times \phi(\vec{P_2}) = \phi(\vec{P_1} \times \vec{P_2})$$

This is how you do the above with a quaternion:

$$\vec{P'} = q P q^{-1}$$

which is equivalent to:

$$R_q = \begin{bmatrix} 1-2y^2 - 2 z^2 & 2xy-2wz & 2xz+2wy \\ 2xy+2wz & 1-2x^2-2z^2 & 2yz-2wx \\ 2xz-2wy & 2yz+2wx & 1-2x^2-2y^2 \end{bmatrix}$$

Spherical Linear Interpolation

$$q(t) = \frac{sin \theta (1-t)}{sin \theta} q_1 + \frac{sin \theta t}{sin \theta} q_2$$ where $0 \le t \le 1$