Thanks very much for pointing this out. I would like to work out some
generally accepted terminology for these transforms, I had a look at some
other websites and have come up with the following definitions, do these
look right to you?
Orthogonal transform - can represent rotations only. positive determinant.
Affine transform - can represent translations, rotations, scaling (may
be different in x,y and z dimensions), shearing and reflections. Lines
remain straight and parallel lines remain parallel, but the angle between
intersecting lines can change.
Euclidean transformations - transformations preserve both angles and lengths
(ie translations, rotations and reflections). Determinant negative if
there is a reflection.
Rigid transformation - transformation which can represent the movement
of a solid object (ie translations and rotations only).
Special Affine transform - an affine transform where determinant of 3x3
part equals unity.