Affine/Linear Transformation
- Most used deformations \(p'=f(p)\)
-
- can be represented by a matrix \(\rightarrow\) can be sent as uniform parameter to the shader.
- - In standard 3D coordinates \(p=(x,y,z)\)
-
- \(p'=\mathrm{L}\,p+t\)
-
- \(\mathrm{L}\): linear component (3\(\times\)3 matrix)
- \(t\): translation
- - In homogeneous coordinates \(p=(x,y,z,1)\)
-
- \(p'=\mathrm{A}\,p\)
-
- \(\mathrm{A}\): 4 \(\times\) 4 matrix
- Properties:
-
- - \(\mathrm{det}(\mathrm{L})\): Change of volume when applying \(L\) to a shape
- - \(\mathrm{L}^{T}\,\mathrm{L}=1\) \(\Rightarrow\) Isometry
\(\mathrm{A}=\left(\begin{array}{c|c} \mathrm{L} & t \\ \hline 0 & 1\end{array} \right)= \scriptsize \left(\begin{array}{ccc|c} \mathrm{L}_{xx} &
\mathrm{L}_{xy} & \mathrm{L}_{xz} & t_x \\ \mathrm{L}_{yx} & \mathrm{L}_{yy} & \mathrm{L}_{yz} & t_y \\ \mathrm{L}_{zx} & \mathrm{L}_{zy} & \mathrm{L}_{zz}
& t_z \\ \hline 0 & 0 & 0 & 1 \end{array} \right)\)