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Nehushtan

Member

Registered: 2013-03-09

Posts: 957

The isometry group of the plane

 

 

 

 

So what possible transformations are isometries of the plane? You might think that there were a whole bunch of them: rotations, reflections, and translations. (A reflection followed by a translation is sometimes called a glide reflection.) In fact the picture is simpler than that: it turns out that rotations and translations can be built up from reflections alone! A translation in a certain direction is simply a reflection in two axes perpendicular that direction, while a rotation about a point O is a reflection in two axes through O.

 

 

 

  Hence any translation is a reflection in two axes perpendicular to the direction of translation whose distance apart is half the distance to be translated.

 

 

 

 

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Hence any rotation is a reflection in two axes through the centre of rotation whose angular separation is half the angle to be rotated through.

Last edited by Nehushtan (2017-11-20 05:47:30)

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