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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos{\theta} \, \vec{q} + \sin{\theta} \, \vec{q}^\perp}
5. Add the origin point to get the final position.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{P} = \cos{\theta} \, \vec{q} + \sin{\theta} \, \vec{q}^\perp + O}

Shearing

Point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} and unit vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} .

Given point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} to shear by amount Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} ,

1. compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q} = P - O}
2. set up a coordinate system by finding the 2D cross product of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}^\perp = \left\langle -v_y, v_x \right\rangle}
3. project Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}} onto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}^\perp} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\parallel = (\vec{v} \cdot \vec{q}) \vec{v}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\perp = (\vec{v}^\perp \cdot \vec{q}) \, \vec{v}^\perp} , respectively.
4. the shearing vector scales Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\parallel} (in the direction of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} depending on the magnitude of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\perp} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\perp + \alpha \, \left| \vec{q}^\perp \right| \, \vec{v}}
5. Add Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{q}^\parallel} to get the final position

Plugging in definitions and rearranging gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p} = M \, p + t}