CSCE 441 Lecture 9

« previous | Monday, February 3, 2014 | next »

Affine Transformations as Matrices

This material will be on the midterm!

$\left[\begin{array}{c}{\hat{p}}_x\\ {\hat{p}}_y\\ 1\end{array}\right]=\left[\begin{array}{ccc}{M}_{11}& M_{12}& t_x\\ M_{21}& M_{22}& t_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\overrightarrow{p}}_x\\ {\overrightarrow{p}}_y\\ 0\end{array}\right]$

Goal is to construct matrix such that a $2\times 2$ matrix is multiplied by our input vector and a 2-coordinate vector $t$ that is added to our input vector.

Dot Product

$\overrightarrow{v}\cdot \overrightarrow{w}={\overrightarrow{v}}^T\overrightarrow{w}=\left[\begin{array}{cc}{\overrightarrow{v}}_x& {\overrightarrow{v}}_y\end{array}\right]\left[\begin{array}{c}{\overrightarrow{w}}_x\\ {\overrightarrow{w}}_y\end{array}\right]$

2D Cross Product

Multiply by matrix $\perp =\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$

Translation

Our 2 × 2 matrix $M$ is just the identity matrix, and our $t$-coordinates become our translation values:

$\left[\begin{array}{cc}I& t\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]=\left[\begin{array}{c}\overrightarrow{p}+t\\ 1\end{array}\right]$

In expanded notation:

$\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\overrightarrow{p}}_x\\ {\overrightarrow{p}}_y\\ 1\end{array}\right]=\left[\begin{array}{c}{\overrightarrow{p}}_x+t_x\\ {\overrightarrow{p}}_y+t_y\\ 1\end{array}\right]$

Uniform Scaling

$\left[\begin{array}{cc}\alpha I& (1-\alpha )o\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]=\left[\begin{array}{c}\alpha \overrightarrow{p}+(1-\alpha )o\\ 1\end{array}\right]$

Non-Uniform Scaling

$\left[\begin{array}{cc}I+(\alpha -1)\overrightarrow{v}{\overrightarrow{v}}^T& (1-\alpha )\overrightarrow{v}{\overrightarrow{v}}^{T}o\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]$

Rotation

$\left[\begin{array}{cc}I& o\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\cos \theta I+\sin \theta \perp & 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}I& -o\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]$

Shear

$\left[\begin{array}{cc}I+\alpha \overrightarrow{v}({\overrightarrow{v}}^{\perp }{)}^T& -\alpha \overrightarrow{v}{\left({\overrightarrow{v}}^{\perp }\right)}^{T}o\\ 0& 1\end{array}\right]\left[\begin{array}{c}\overrightarrow{p}\\ 1\end{array}\right]$

Finding Affine Transformations

Image of 3 points determines unique affine transformation

$M\left[\begin{array}{ccc}\overrightarrow{p}& \overrightarrow{q}& \overrightarrow{r}\\ 1& 1& 1\end{array}\right]=\left[\begin{array}{ccc}\hat{p}& \hat{q}& \hat{r}\\ 1& 1& 1\end{array}\right]$

Hence

$M=\left[\begin{array}{ccc}\hat{p}& \hat{q}& \hat{r}\\ 1& 1& 1\end{array}\right]{\left[\begin{array}{ccc}\overrightarrow{p}& \overrightarrow{q}& \overrightarrow{r}\\ 1& 1& 1\end{array}\right]}^{-1}$

Composing Transformations

Multiply all of the matrices together

Fractals and Iterated Affine Transformations

Fractals are recursion made visible:

A self-similar shape created from a set of contractive transformations