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Translation

\(t(p)=(x+t_x,y+t_y,z+t_z)\)

\(\mathrm{T} = \left(\begin{array}{llll} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{array}\right) \)

Scaling

\(s(p)=(s_x\,x, s_y\,y, s_z\,z)\)

\(\mathrm{S} = \left(\begin{array}{llll} s_x & 0 & 0 & 0 \\ 0 & s_y & 0 & 0 \\ 0 & 0 & s_z & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \)

Rotation

Several possible representations (see later)

Note: Isometry

Shearing

\(sh_{xy}(p)=(x+\lambda y, y, z)\)
\(sh_{xz}(p)=(x+\lambda z, y, z)\)
\(sh_{yx}(p)=(x, y+\lambda x, z)\)
...

\(\mathrm{Sh}_{xy} = \left(\begin{array}{llll} 1 & \lambda & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \) \(\mathrm{Sh}_{xz} = \left(\begin{array}{llll} 1 & 0 & \lambda & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \) \(\mathrm{Sh}_{yx} = \left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ \lambda & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \) ...

Usually avoided in Graphics

Note: \(\mathrm{det}(\mathrm{Sh})=1 \to \) constant volume (but no isometry)

Affine/Linear Deformations

Translation

Scaling

Rotation

Shearing