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 *E*^{3} 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 *E*^{3}, acts as a translation by [*a b c*]^{T} on *E*^{3}.

• 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 ...

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