# 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.

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 (*)).