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Reflections in the symmetry group 2H_2 of a symmetric bar (dumbbell)

The group 2H_2 has four distinct (oriented) reflections x -> x' = rxr, rr=1.
Each reflection is characterized by the unit direction vector r of the line of reflection.
These reflections are oriented, because each line of reflection has two possible opposite directions,
characterized by the sign of the unit direction vector r.

We have:

r = a

r = -a

r = b

r = -b

Each reflection can be visualized by dragging the red vector perpendicular to the line of reflection
to a particular value of r. It is easily verified that any other choice of r, not in the above list, will
not produce a reflection symmetry of the symmetric bar. The combination of two such reflections at r and r' will yield an
(oriented) rotation y->R~yR=(r'r)y(rr')
from the dicyclic group 2C_2.

[ 2D point groups | GA with Cinderella ]

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