It has already been demonstrated, in Section 1.4, that point coordinates and vector coefficients display a different behaviour when the coordinate origin is shifted. The same happens when a translation is applied to a pair of points. The coordinates of the points will be changed according to
However, the distance between the points will be invariant:
Distances are absolute values of vectors, see Section 1.6. Usually point coordinates and vector coefficients are described by the same kind of columns and are difficult to distinguish. It is a great advantage of the augmented columns to provide a clear distinction between these quantities.
If
and
are the augmented columns of coordinates of the
points and ,
Let T be a translation, (I,t) its matrix-column pair,
its augmented matrix, r the column of
coefficients of the distance vector r between and , and
the augmented column of r. Then,
When using augmented columns and matrices, the coefficients of t are multiplied with the last coefficient 0 of the column and thus become ineffective.
This behaviour is valid not only for translations but holds in general
for affine mappings, and thus for isometries and crystallographic symmetry
operations:
Whereas point coordinates are transformed by
,
vector coefficients r are affected only by the matrix part W:
Note that is different from . The latter expression describes the image point of the point with the coordinates .
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