My task is to transform the cube from the left corner to the big cube in the middle:
What I did was:
First i scale the cube:
$$ \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} $$
Next i rotate it $45^\circ$:
$$ \begin{pmatrix} \cos(45^\circ) & -\sin(45^\circ) & 0 \\ \sin(45^\circ) & \cos(45^\circ) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} $$
And last i translate the cube:
$$ \begin{pmatrix} 1 & 0 & 10 \\ 0 & 1 & 7 \\ 0 & 0 & 1 \\ \end{pmatrix} $$
When i combine this matrices i get my transformation matrix:
$$ \begin{pmatrix} 4\cos(45^\circ) & -4\sin(45^\circ) & 40\cos(45^\circ) - 28\sin(45^\circ) \\ 4\sin(45^\circ) & 4\cos(45^\circ) & 40\cos(45^\circ) + 28\sin(45^\circ) \\ 0 & 0 & 1 \\ \end{pmatrix} $$
But then i tried to apply this matrix to the point $(0,1,1)$ and i got an wrong result:
$$(5.6, 50.9116 ,1)$$
As you can see it should be around:
$$(7.1, 9.9, 1)$$
What did I wrong? Thanks!
There is a mistake in your multiplication.
$$ T_{total}=\begin{pmatrix} 4\cos(45^\circ) & -4\sin(45^\circ) & 40\cos(45^\circ) - 28\sin(45^\circ) \\ 4\sin(45^\circ) & 4\cos(45^\circ) & \color{red}{28\cos(45^\circ) + 40\sin(45^\circ)} \\ 0 & 0 & 1 \\ \end{pmatrix} $$
But, since $sin (45^\circ)$ and $cos (45^\circ)$ have the same value, the problem is not from this point.
The problem is related to the order of the Transforms. In matrix multiplication, you should care about matrix order.
$T1$ is scale transform
$T2$ is rotation transform
$T3$ is translation transform
you computed:
$$X_{new}=(T1 \times T2 \times T3) \times X_{old}$$
while, you need to scale first then rotation then translation:
$$X_{new}=(T3 \times T2 \times T1) \times X_{old}$$
$$T3 \times T2 \times T1=\begin{pmatrix} 4\cos(45^\circ) & -4\sin(45^\circ) & 10 \\ 4\sin(45^\circ) & 4\cos(45^\circ) & 7 \\ 0 & 0 & 1 \\ \end{pmatrix}$$
$$T1\times T2 \times T3 \times \begin{pmatrix} 0 \\ 1 \\ 1 \\ \end{pmatrix}=\begin{pmatrix} 5.66 \\ 50.91 \\ 1 \\ \end{pmatrix}$$
While
$$T3\times T2 \times T1 \times \begin{pmatrix} 0 \\ 1 \\ 1 \\ \end{pmatrix}=\begin{pmatrix} 7.17 \\ 9.83 \\ 1 \\ \end{pmatrix}$$