Thread: Isometries

I'm looking for a simple test for telling whether a function f is a translation, rotation, or glide reflection from the positions of f(A), f(B), and f(C).

Thanks!

Hello meggnog

Originally Posted by meggnog

I'm looking for a simple test for telling whether a function f is a translation, rotation, or glide reflection from the positions of f(A), f(B), and f(C).

Thanks!

I assume from your question that you are given three (non-collinear) points A, B, C on the original object, and their images f(A), f(B), f(C). Then the process is something like:

1. Look first at the sense in which the original points and their images are ordered: in other words, whether they are arranged in clockwise or anticlockwise order in the object and its image. If they are in opposite senses (i.e one is clockwise and the other anti-clockwise), then the isometry is a reflection or a glide reflection.
   
   To distinguish between a reflection and a glide reflection, join pairs of corresponding points: A to f(A), etc. If all three of these lines are parallel, then it's a simple reflection; if not, it's a glide reflection.
   .
2. If, conversely, the sense of the points and their images are the same (i.e. both clockwise or both anti-clockwise) then it's either a translation or a rotation.
   
   To distinguish between them, again you can join pairs of corresponding points. If all three lines are parallel, then it's a translation; if not, it's a rotation.

Grandad