Affine Transformation

The transformations of this form are called affine if the coordinates are linear functions of the coordinates

(1.2)

where are constants. Above equations have nontrivial solution for , so that the determinant

(1.3)

Properties:

(a) Inverse affine transformation is also affine, because by solving 1.2 one obtains in terms of :

(1.4)

(b) Points lying on a plane before transformation lies on a different plane .
Say is the equation of plane , then after the transformation they lie on a different plane , which is the equation of plane .

(c) Points lying on a straight line, which is the intersection of two different planes and , will now lie again on a straight line, which is basically the intersection of two planes and . Hence, any straight segment is transformed into a straight segment and any vector to a vector.