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
|TVT-V = I,|
where I is the identity transform. If V is a non-zero vector the for no point
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:
On the other hand, if S' is known to be a translated image of S, then for any points P and Q in S,
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.
|What if applet does not run?|
The following observations are noteworthy:
Translation preserves the orientation. For example, if a polygon is traversed clockwise, its translated image is likewise traversed clockwise.
Translation is isometry: a translation preserves distances.
Translation preserves angles.
Translation maps parallel lines onto parallel lines and, moreover, a line and its image are also parallel.
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.
Successive translations result in a translation. Moreover,
TVTU = TV+U.
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:
- A Property of Rhombi
- A Property of the Line IO
- Bottles in a Wine Rack
- Building a Bridge
- Building Bridges
- Circles with Equal Collinear Chords
- Find a Common Chord of Given Length
- Join Circles by Given Segment
- Translated Triangles
- Plane Isometries
- Plane Isometries As Complex Functions
- Affine Transformations (definition)
- Projective transformation (definition)
Copyright © 1996-2018 Alexander Bogomolny