CSCE 441 Lecture 9

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Affine Transformations as Matrices

This material will be on the midterm!

${\begin{bmatrix}{\hat {p}}_{x}\\{\hat {p}}_{y}\\1\end{bmatrix}}={\begin{bmatrix}M_{11}&M_{12}&t_{x}\\M_{21}&M_{22}&t_{y}\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}_{x}\\{\vec {p}}_{y}\\0\end{bmatrix}}$

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

${\vec {v}}\cdot {\vec {w}}={\vec {v}}^{T}\,{\vec {w}}={\begin{bmatrix}{\vec {v}}_{x}&{\vec {v}}_{y}\end{bmatrix}}\,{\begin{bmatrix}{\vec {w}}_{x}\\{\vec {w}}_{y}\end{bmatrix}}$

2D Cross Product

Multiply by matrix $\perp ={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

Translation

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

${\begin{bmatrix}I&t\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}={\begin{bmatrix}{\vec {p}}+t\\1\end{bmatrix}}$

In expanded notation:

${\begin{bmatrix}1&0&t_{x}\\0&1&t_{y}\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}_{x}\\{\vec {p}}_{y}\\1\end{bmatrix}}={\begin{bmatrix}{\vec {p}}_{x}+t_{x}\\{\vec {p}}_{y}+t_{y}\\1\end{bmatrix}}$

Uniform Scaling

${\begin{bmatrix}\alpha \,I&\left(1-\alpha \right)\,o\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}={\begin{bmatrix}\alpha \,{\vec {p}}+(1-\alpha )\,o\\1\end{bmatrix}}$

Non-Uniform Scaling

${\begin{bmatrix}I+\left(\alpha -1\right)\,{\vec {v}}\,{\vec {v}}^{T}&\left(1-\alpha \right)\,{\vec {v}}\,{\vec {v}}^{T}\,o\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}$

Rotation

${\begin{bmatrix}I&o\\0&1\end{bmatrix}}\,{\begin{bmatrix}\cos {\theta }\,I+\sin {\theta }\,\perp &0\\0&1\end{bmatrix}}\,{\begin{bmatrix}I&-o\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}$

Shear

${\begin{bmatrix}I+\alpha \,{\vec {v}}\,({\vec {v}}^{\perp })^{T}&-\alpha \,{\vec {v}}\left({\vec {v}}^{\perp }\right)^{T}\,o\\0&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}\\1\end{bmatrix}}$

Finding Affine Transformations

Image of 3 points determines unique affine transformation

$M\,{\begin{bmatrix}{\vec {p}}&{\vec {q}}&{\vec {r}}\\1&1&1\end{bmatrix}}={\begin{bmatrix}{\hat {p}}&{\hat {q}}&{\hat {r}}\\1&1&1\end{bmatrix}}$

Hence

$M={\begin{bmatrix}{\hat {p}}&{\hat {q}}&{\hat {r}}\\1&1&1\end{bmatrix}}\,{\begin{bmatrix}{\vec {p}}&{\vec {q}}&{\vec {r}}\\1&1&1\end{bmatrix}}^{-1}$

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