CSCE 441 Lecture 8

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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

$v^{\perp }=\left\langle -v_{y},v_{x}\right\rangle$

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 ${\vec {t}}$ to point $P$:

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

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 ${\vec {v}}$ (assumed unit length)

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

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

Substituting definitions and rearranging the formula gives

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

Rotation

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

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

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

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

Shearing

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

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

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

Plugging in definitions and rearranging gives

${\hat {P}}=P+\alpha \left({\vec {v}}^{\perp }\cdot {\vec {q}}\right)\,{\vec {v}}$

Transformations as Matrices

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

${\hat {p}}=M\,p+t$