Scaling

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Scaling shifts coordinates similar to translation. The difference is, that translation is independent of the current coordinates. Scaling however shifts the coordinates depending on the current values. So the same trnasformation matrix has various effects on different vectors.

For the sake of simplicity scaling is explained for two-dimensional space. But the behavior of the parameters is the same for an additional third dimension. Scaling is achieved by modifying the components $a$ and $d$ on the first diagonal of the transformation matrix $\mathbf{T}$, while the other components are zero:

$\mathbf{T}_s= \left[\begin{array}{cc} a & 0\\ 0 & d \end{array}\right]$

The computation of scaling looks as follows in component notation:

$\vec{\mathbf{q}}_1=\mathbf{T}_s\cdot\vec{\mathbf{q}}_0= \left[\begin{array}{cc} a & 0\\ 0 & d \end{array}\right]\cdot \left[\begin{array}{c} x_0\\ y_0 \end{array}\right]= \left[\begin{array}{c} a\cdot x_0\\ d\cdot y_0 \end{array}\right]$

So the values $a$ and $d$ act as factors for the x- and the y-component of the scaled vector. Several different cases $a$ and $d$ can be considered, for example:

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$\mathbf{T}_0= \left[\begin{array}{cc} a=1 & 0\\ 0 & d=1 \end{array}\right]$

$\mathbf{T}_0\cdot \vec{\mathbf{q}}_0 = \vec{\mathbf{q}}_0$

$\mathbf{T}$ is the identity matrix → no effect

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$\mathbf{T}_1= \left[\begin{array}{cc} a>1 & 0\\ 0 & d=1 \end{array}\right]$

$\mathbf{T}_1\cdot \vec{\mathbf{q}}_0 = \vec{\mathbf{q}}_1$

x-component is scaled up → vector shifted to positive x-direction