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 TV(P) = P'. If P' = TV(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,

  TVT-V = I,

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

The translation transform TV 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' = TV(S). To determine TV(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.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

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.

Here is a list of several problems that are solved with the help of translation:

  1. A Property of Rhombi
  2. A Property of the Line IO
  3. Bottles in a Wine Rack
  4. Building a Bridge
  5. Building Bridges
  6. Circles with Equal Collinear Chords
  7. Find a Common Chord of Given Length
  8. Join Circles by Given Segment
  9. Translated Triangles

Geometric Transformations

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny