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.
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
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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