3.4 Affine and Projective Transformation

Transform Name	Matrix Form	Degrees of Freedom	Degrees of Freedom
Euclidean	$\left[ {\begin{array}{*{20}{c}} \pmb{R} & \pmb{t}\\ { {\pmb{0}^T}}&1 \end{array}} \right]$	6	Length, angle, volume
Similarity	$\left[ {\begin{array}{*{20}{c}} {s \pmb{R}}& \pmb{t}\\ { { \pmb{0}^T}}&1 \end{array}} \right]$	7	volume ratio
Affine	$\left[ {\begin{array}{*{20}{c}} \pmb{A} & \pmb{t}\\ { {\pmb{0}^T}} & 1 \end{array}} \right]$	12	Parallelism, volume ratio
Perspective	$\left[ {\begin{array}{*{20}{c}} \pmb{A} & \pmb{t}\\ { {\pmb{a}^T}} & v \end{array}} \right]$	15	Plane intersection and tangency