CSCE 441 Lecture 37

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Quaternions and Complex Numbers

how they should be used.

Complex Numbers

Defined by a real and imaginary part:

$a+bi$, where $i=\sqrt{-1}$.

Simple operations:

• $(a+bi)+(c+di)=(a+c)+(b+d)i$
• $⟨a+bi⟩(c+di)=ac-bd+(ad+bc)i$
• Conjugation: $z=a+bi$ becomes $\overline{z}=a-bi$. What's impressive about this is that $z\overline{z}=a^2+b^2={\left|z\right|}^2$, where $\left|z\right|=\sqrt{a^2+b^2}$

Relation to Graphics

Given a point $(x,y)$, rotate that point about the origin by $\theta$:

$\left[\begin{array}{c}\hat{x}\\ \hat{y}\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

Represent $(x,y)$ by a complex number: $(x+yi)$

$(x+yy)(\cos \theta +i\sin \theta )=(x\cos \theta -y\sin \theta )+(x\sin \theta +y\cos \theta )i$

Rotation is just multiplication by a complex number.

Quaternions

Sir William Rowan Hamilton attempted to extend complex numbers from 2D to 3D, but this is now provably impossible. He discovered a generalization to 4D and wrote it on the side of a bridge in Dublin.

One real part, 3 complex parts:

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

From this, we get

• $ij=k$
• $ji=-k$
• $jk=i$
• $kj=-i$
• $ki=j$
• $ik=-j$

$q=(s,\overrightarrow{v})=s+v_{x}i+v_{y}j+v_{z}k$

We can define an algebra on $\mathbb{ℍ}$

$q_1=(s_1,{\overrightarrow{v}}_1)$ and $q_2=(s_2,{\overrightarrow{v}}_2)$

• Multiplication: $q_1{q}_2=(s_1{s}_2-v_1\cdot v_2,s_1{\overrightarrow{v}}_2+s_2{\overrightarrow{v}}_1+{\overrightarrow{v}}_1\times {\overrightarrow{v}}_2)$
  • Order matters in quaternion multiplication!
• Conjugation: $\overrightarrow{q}=(s-\overrightarrow{v})$, and $⟨q\overline{s}⟩=s^2+{\left|v\right|}^2={\left|q\right|}^2$
• Inversion: $q^{-1}=\frac{\overline{q}}{\left|q\right|}$

Relation to Graphics

Claim: unit quaternions represent 3D rotation

Convert from 3D to 4D: $p=(0,\overrightarrow{r})$, where $\overrightarrow{r}=⟨x,y,z⟩$.

$q=(\cos \theta ,\sin \theta \overrightarrow{v})$, where $\left|\overrightarrow{v}\right|=1$.

let $\overrightarrow{v}$ be parallel to the axis of rotation in 3D.

In this case, $pq=qp=(-\sin \theta ,\cos \theta \overrightarrow{v})$. This gives a nonzero real component... uh oh, we now have a 4D number, not a 3D number.

let ${\overrightarrow{v}}^{\perp }$ be a vector component in the plane normal to $\overrightarrow{r}$:

$qp$ represents a positive (ccw) rotation, and $pq$ represents a negative (cw) rotation.

Computing $qp\overline{q}$ rotates the component of $\overrightarrow{v}$ perpendicular to $r$ by $2\theta$ and leaves the parallel component alone.

Thus the quaternion representing rotation about the unit axis $\overrightarrow{v}$ by $\theta$ is $q(0,\overrightarrow{r})\overline{q}$, where $q=(\cos \left(\frac{\theta }{2}\right),\sin \left(\frac{\theta }{2}\right)\overrightarrow{v})$

Quaternions vs. Matrices

This seems like magic... This is perhaps the wrong use for quaternions

• Quaternions take less space (4 vs. 9)
• Rotating a vector requires 28 multiplications using quaternions vs. 9 for matrices
• Composing two rotations using quaternions $q_1{q}_2$ requires 16 multiplications vs. 27 for matrices
• Quaternions are not hardware-accelerated whereas matrices are.

Quaternions and Interpolation

Unit quaternions represent points on a 4D hypersphere

Interpolation on the sphere gives rotations that bend the least.

Recall we used SLERP to interpolate vectors on the surface of a 3D sphere.

May need to interpolate between $q_1$ and $-q_2$.

$q(t)=\frac{\sin (\theta (1-t))q_1+\sin (\theta t)q_2}{\sin \theta }$, where ${\cos }^{-1}(q_1\cdot q_2)=\theta$.

Other Graphics Uses for Quaternions

Skeletal animation: moving bones usually requires a lot of composed rotations.