Translation Transform

Given a vector V and a point P. A vector can be specified by its direction and length. Translation (often a parallel translation) P' of P by V is the point such that PP' equals the vector V. In other words, P' is located at the distance from P equal to the length of V and in the same direction.

P' exists for any P. Let's write T_V(P) = P'. If P' = T_V(P), then P is the translation of P' by -V: P = T_-V(P'), where -V is the vector of the same length as V, but pointing in the opposite direction. Formally,

where I is the identity transform. If V is a non-zero vector the for no point T_V(P) = P, i.e., a nontrivial translation has no fixed points.

The translation transform T_V applies to arbitrary shapes point-by-point. Each point of a given shape S is translated by V, and the collection of these translations is the translated image of S: S' = T_V(S). To determine T_V(S) when S is a polygon, suffice it to translate its vertices. This is exactly what has been done in the applet below.

On the other hand, if S' is known to be a translated image of S, then for any points P and Q in S, PP' = QQ'. Therefore, the translation that maps S on S' is uniquely determined by any pair of points P/P'.

In the applet, you can create polygons with a desired number of vertices, drag the vertices one at a time, or drag the polygon as a whole. Vectors of translations can also be dragged. They rotate if dragged near their endpoints, or translate if dragged nearer their midpoint.

The following observations are noteworthy:

1. Translation preserves the orientation. For example, if a polygon is traversed clockwise, its translated image is likewise traversed clockwise.

2. Translation is isometry: a translation preserves distances.

3. Translation preserves angles.

4. Translation maps parallel lines onto parallel lines and, moreover, a line and its image are also parallel.

5. Except for the trivial translation by a zero vector, translation have no fixed points. All lines parallel to the vector of a translation are fixed by the translation.

6. Successive translations result in a translation. Moreover,

   TVTU = TV+U.

   
   

7. The order of translations does not matter: any two translations commute. In fact, for two vectors U and V,

   TUTV = TVTU.

Copyright © 1996-2008 Alexander Bogomolny