Transformations in 3-space are in many ways analogous to those in 2-space.
• Translations can be incorporated by treating three-dimensional space as the subset E3 defined by w = 1 in the four-dimensional space of points (x, y, z, w). A linear transformation whose matrix has the form , when restricted to E3, acts as a translation by [a b c]T on E3.
• If T is any continuous transformation that takes lines to lines, and O denotes the origin of 3-space, then we can define
and the result ...