Parallel displacement

A special case of a motion in which all points of the space are transferred in one and the same direction (along a line in that space) over one and the same distance. In other words, if $ M $ is the original and $ M ^ \prime $ is the shifted position of a point, then the vector $ {MM ^ \prime } vec $ is one and the same for all pairs of points corresponding to each other under the given transformation.

In the plane a parallel displacement may analytically be expressed in a rectangular coordinate system $ ( x, y) $ by the formulas

$$ \tag{* } \widetilde{x} = x+ a,\ \ \widetilde{y} = y+ b, $$

where $ {MM ^ \prime } vec = ( a, b) $.

The collection of all parallel displacements forms a group, which in a Euclidean space is a subgroup of the group of motions, and in an affine space is a subgroup of the group of affine transformations.

Comments

In the absolute plane a parallel displacement (cf. Absolute geometry) is the product of reflections in two parallel lines (cf. Reflection). Thus, in Euclidean geometry a parallel displacement is a translation, expressible in affine (or Cartesian) coordinates by $ ( x, y) \rightarrow ( \widetilde{x} , \widetilde{y} ) $( see (*)).

But in Lobachevskii geometry a parallel displacement moves each point along a horocycle, whereas a translation is different, namely the product of half-turns about two distinct points, or the product of reflections in two lines that have a common perpendicular.

References

[a1]	H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 224–240