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#1 2016-04-22 14:36:50

Nehushtan
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Registered: 2013-03-09
Posts: 894
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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 reflections and translations can be built up from rotations 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.

 
 
 
 

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.


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