transformation-math.md

Transformation Math

Augmented Matrix

Using Augmented Matrix to represent $\mathbf{y}=A\mathbf{x}+\mathbf{b}$ as follows,

$$\begin{bmatrix} \mathbf{y} \\ 1 \end{bmatrix} = \left[\begin{matrix} &A& \\ 0&\cdots&0 \end{matrix} \mkern 0.5em \bigg\rvert \mkern 0.5em \begin{matrix} \mathbf{b} \\ 1 \end{matrix}\right]\begin{bmatrix} \mathbf{x} \\ 1 \end{bmatrix}$$

Or equivalently as,

$$\widehat{\mathbf{y}} = M\widehat{\mathbf{x}} $$

This exercise will cover only affine transformations. The augmented representation, however, is equally capable to represent projective transformations as well.

Translation

Preserves distances and oriented angles

$$M_t\left(\begin{matrix}h\\ k\end{matrix}\right) = \begin{bmatrix} 1&0&h \\ 0&1&k \\ 0&0&1 \end{bmatrix}$$

Rotations

Preserves relative angles and distances

$$M_r\left(\theta\right) = \begin{bmatrix} \cos\theta&-\sin\theta&0 \\ \sin\theta&\cos\theta&0 \\ 0&0&1 \end{bmatrix}$$

Uniform Scaling

Preserves relative angles and ratio between distances

$$M_{u}\left(\lambda\right) = \begin{bmatrix} \lambda&0&0 \\ 0&\lambda&0 \\ 0&0&1 \end{bmatrix}$$

Non-uniform Scaling and Shear

Preserves parallelism

$$M_{s}\left(\begin{matrix}s_x\\ s_y\end{matrix}\right) = \begin{bmatrix} s_x&0&0 \\ 0&s_y&0 \\ 0&0&1 \end{bmatrix}$$

$$M_{\mathrm{sh}}\left(\begin{matrix}h_x\\ h_y\end{matrix}\right) = \begin{bmatrix} 1&h_y&0 \\ h_x&1&0 \\ 0&0&1 \end{bmatrix}$$