Linear transformations

A rotation is a special case of a linear transformation, which is generally expressed by an $n \times n$ matrix, $M$, assuming the transformations are performed over ${\mathbb{R}}^n$. Consider transforming a point $(x,y)$ in a 2D robot, ${\cal A}$, as

$$\begin{pmatrix}m_{11} & m_{12} m_{21} & m_{22} \end{pmatrix} \begin{pmatrix}x y \end{pmatrix}.$$ (3.82)

If $M$ is a rotation matrix, then the size and shape of ${\cal A}$ will remain the same. In some applications, however, it may be desirable to distort these. The robot can be scaled by $m_{11}$ along the $x$-axis and $m_{22}$ along the $y$-axis by applying

$$\begin{pmatrix}m_{11} & 0 0 & m_{22} \end{pmatrix} \begin{pmatrix}x y \end{pmatrix},$$ (3.83)

for positive real values $m_{11}$ and $m_{22}$. If one of them is negated, then a mirror image of ${\cal A}$ is obtained. In addition to scaling, ${\cal A}$ can be sheared by applying

$$\begin{pmatrix}1 & m_{12} 0 & 1 \end{pmatrix} \begin{pmatrix}x y \end{pmatrix}$$ (3.84)

for $m_{12} \not = 0$. The case of $m_{12} = 1$ is shown in Figure 3.28.

Figure 3.28: Shearing transformations may be performed.

The scaling, shearing, and rotation matrices may be multiplied together to yield a general transformation matrix that explicitly parameterizes each effect. It is also possible to extend the $M$ from $n \times n$ to $(n+1) \times (n+1)$ to obtain a homogeneous transformation matrix that includes translation. Also, the concepts extend in a straightforward way to ${\mathbb{R}}^3$ and beyond. This enables the additional effects of scaling and shearing to be incorporated directly into the concepts from Sections 3.2-3.4.