CSCE 441 Lecture 8

« previous | Friday, January 31, 2014 | next »

2D Transformations

Depends on choice of origin (0,0) and axes.

Analogous to assembly language:

• Assembly is a necessary evil for computation
• But we'd like to think at a higher level

Vector operations

dot product

$v\cdot w=\left|v\right|\left|w\right|\cos \theta =v_x{w}_x+v_y{w}_y$

commutative

$v\cdot v={\left|v\right|}^2$

2D cross product

$v^{\perp }=⟨-v_y,v_x⟩$

Rotates $v$ by $\frac{\pi }{2}$

Types of Transformations

1. Conformal
   • Preserves angles
   • Translation, rotation, uniform scaling
2. Affine
   • Represented by matrix multiplication
   • Translation, rotation, general scaling, shear

Translation

Add vector $\overrightarrow{t}$ to point $P$:

$\hat{P}=P+\overrightarrow{t}$

Uniform Scaling

origin $O$ about which to scale

Given point $P$ to scale and scaling factor $\alpha$,

1. find vector from $O$ to $P$: $P-O$
2. scale that vector by alpha: $\alpha (P-O)$
3. add the vector back to $O$ to get final position: $O+\alpha (P-O)$

Rearranging the terms gives

$\hat{P}=(1-\alpha )O+\alpha P$

Non-Uniform Scaling

Point and vector:

• origin point $O$ for center of scaling
• direction of scale $\overrightarrow{v}$ (assumed unit length)

Given point $P$ to scale and a scaling factor $\alpha$,

1. find vector from $O$ to $P$: $\overrightarrow{q}=P-O$
2. take vector projection of $\overrightarrow{q}$ onto $\overrightarrow{v}$: ${\overrightarrow{v}}^{\parallel }={\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_{\overrightarrow{v}}(\overrightarrow{q})=(\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}$
3. use projection to find residual vector: ${\overrightarrow{v}}^{\perp }=\overrightarrow{q}-{\overrightarrow{q}}^{\parallel }=\overrightarrow{q}-(\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}$
4. scale the $\overrightarrow{v}$-parallel component of $\overrightarrow{q}$: $\alpha {\overrightarrow{q}}^{\parallel }=\alpha (\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}$
5. add residual vector back in and add to origin point $O$ to get final point: $O+{\overrightarrow{q}}^{\perp }+\alpha {\overrightarrow{q}}^{\parallel }=O+\overrightarrow{q}-(\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}+\alpha (\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}$

Substituting definitions and rearranging the formula gives

$\hat{P}=P+(\alpha -1)(\overrightarrow{v}\cdot (P-O))\overrightarrow{v}$

Rotation

Need center of rotation $O$ and angle $\theta$

Given point $P$ to rotate about point $O$,

1. find vector from $O$ to $P$: $\overrightarrow{q}=P-O$
2. set up a "coordinate system" by taking the 2D cross product ${\overrightarrow{q}}^{\perp }=⟨-q_y,q_x⟩$
3. using $\overrightarrow{q}$ and ${\overrightarrow{q}}^{\perp }$ as new axes, observe that the rotated point is at position $(\left|\overrightarrow{q}\right|\cos \theta ,\left|\overrightarrow{q}\right|\sin \theta )$
4. hence the vector from $O$ to $\hat{P}$ is given by $\cos \theta \overrightarrow{q}+\sin \theta {\overrightarrow{q}}^{\perp }$
5. Add the origin point to get the final position.

$\hat{P}=\cos \theta \overrightarrow{q}+\sin \theta {\overrightarrow{q}}^{\perp }+O$

Shearing

Point $O$ and unit vector $\overrightarrow{v}$.

Given point $P$ to shear by amount $\alpha$,

1. compute $\overrightarrow{q}=P-O$
2. set up a coordinate system by finding the 2D cross product of $\overrightarrow{v}$: ${\overrightarrow{v}}^{\perp }=⟨-v_y,v_x⟩$
3. project $\overrightarrow{q}$ onto $\overrightarrow{v}$ and ${\overrightarrow{v}}^{\perp }$ to get ${\overrightarrow{q}}^{\parallel }=(\overrightarrow{v}\cdot \overrightarrow{q})\overrightarrow{v}$ and ${\overrightarrow{q}}^{\perp }=({\overrightarrow{v}}^{\perp }\cdot \overrightarrow{q}){\overrightarrow{v}}^{\perp }$, respectively.
4. the shearing vector scales ${\overrightarrow{q}}^{\parallel }$ (in the direction of $\overrightarrow{v}$ depending on the magnitude of ${\overrightarrow{q}}^{\perp }$ and $\alpha$: ${\overrightarrow{q}}^{\perp }+\alpha \left|{\overrightarrow{q}}^{\perp }\right|\overrightarrow{v}$
5. Add ${\overrightarrow{q}}^{\parallel }$ to get the final position

Plugging in definitions and rearranging gives

$\hat{P}=P+\alpha ({\overrightarrow{v}}^{\perp }\cdot \overrightarrow{q})\overrightarrow{v}$

Transformations as Matrices

• Compact representation of all affine transformations
• allows multiple transformatinos to be represented as a single matrix
• requires coordinates

$\hat{p}=Mp+t$