CSCE 441 Lecture 37

« previous | Wednesday, April 23, 2014 | next »

Quaternions and Complex Numbers

how they should be used.

Complex Numbers

Defined by a real and imaginary part:

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

Simple operations:

• $\left(a+b\,i\right)+\left(c+d\,i\right)=(a+c)+(b+d)\,i$
• $\left\langle a+b\,i\right\rangle \,\left(c+d\,i\right)=a\,c-b\,d+\left(a\,d+b\,c\right)\,i$
• Conjugation: $z=a+b\,i$ becomes ${\bar {z}}=a-b\,i$. What's impressive about this is that $z\,{\bar {z}}=a^{2}+b^{2}=\left|z\right|^{2}$, where $\left|z\right|={\sqrt {a^{2}+b^{2}}}$

Relation to Graphics

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

${\begin{bmatrix}{\hat {x}}\\{\hat {y}}\end{bmatrix}}={\begin{bmatrix}\cos {\theta }&-\sin {\theta }\\\sin {\theta }&\cos {\theta }\end{bmatrix}}\,{\begin{bmatrix}x\\y\end{bmatrix}}$

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

$(x+y\,y)\,\left(\cos {\theta }+i\,\sin {\theta }\right)=\left(x\,\cos {\theta }-y\,\sin {\theta }\right)+\left(x\,\sin {\theta }+y\,\cos {\theta }\right)\,i$

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:

$i^{2}=j^{2}=k^{2}=i\,j\,k=-1$

From this, we get

• $i\,j=k$
• $j\,i=-k$
• $j\,k=i$
• $k\,j=-i$
• $k\,i=j$
• $i\,k=-j$

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

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

$q_{1}=\left(s_{1},\,{\vec {v}}_{1}\right)$ and $q_{2}=\left(s_{2},{\vec {v}}_{2}\right)$

• Multiplication: $q_{1}\,q_{2}=\left(s_{1}\,s_{2}-v_{1}\cdot v_{2},s_{1}\,{\vec {v}}_{2}+s_{2}\,{\vec {v}}_{1}+{\vec {v}}_{1}\times {\vec {v}}_{2}\right)$
  • Order matters in quaternion multiplication!
• Conjugation: ${\vec {q}}=\left(s\,-{\vec {v}}\right)$, and $\left\langle q\,{\bar {s}}\right\rangle =s^{2}+\left|v\right|^{2}=\left|q\right|^{2}$
• Inversion: $q^{-1}={\frac {\bar {q}}{\left|q\right|}}$

Relation to Graphics

Claim: unit quaternions represent 3D rotation

Convert from 3D to 4D: $p=\left(0,{\vec {r}}\right)$, where ${\vec {r}}=\left\langle x,y,z\right\rangle$.

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

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

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

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

$q\,p$ represents a positive (ccw) rotation, and $p\,q$ represents a negative (cw) rotation.

Computing $q\,p\,{\bar {q}}$ rotates the component of ${\vec {v}}$ perpendicular to $r$ by $2\theta$ and leaves the parallel component alone.

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

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 $-q_{2}$.

$q(t)={\frac {\sin {\left(\theta \,(1-t)\right)}\,q_{1}+\sin {(\theta \,t)}q_{2}}{\sin {\theta }}}$, where $\cos ^{-1}{\left(q_{1}\cdot q_{2}\right)}=\theta$.

Other Graphics Uses for Quaternions

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