       Next: The matrix-column pairs of crystallographic symmetry Up: The description of mappings by ... Previous: () matrices

## Transformation of vector coefficients

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 , is the augmented column of the coefficients of the distance vector r between and . The last coefficient of is zero, because of . It follows that columns of vector coefficients are augmented in another way than columns of point coordinates.

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, (4.4.1)

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:  (4.4.2)

Whereas point coordinates are transformed by , vector coefficients r are affected only by the matrix part W: (4.4.3)

In other words: if (W,w) describes an affine mapping (isometry, crystallographic symmetry operation) in point space, then W describes the corresponding mapping in vector space. For vector coefficients, the column part w does not contribute to the mapping. This is valid for any vector, e.g., also for the basis vectors of the coordinate system.

Note that is different from . The latter expression describes the image point of the point with the coordinates .     Next: The matrix-column pairs of crystallographic symmetry Up: The description of mappings by ... Previous: () matrices