Reflections in the "symmetry group" 2H_1 of a non-symmetric figure

The group 2H_1 has two distinct (oriented) reflections x -> x' = rxr, rr=1. Each reflection is characterized by a unit direction vector r=a or r=-a=b of the line of reflection. a is the vector from the fixed point P to the object (blue circle) labeled "1". These reflections are oriented, because the line of reflection has two possible opposite directions, characterized by the sign of a.

Both orientations of the 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 choice other than r=a or r=-a=b, will not produce a reflection symmetry of the figure below. The combination of two reflections at r and r' will yield an (oriented) rotation y->y'= R~yR = (r'r)y(rr') from the dicyclic group 2C_1 (with rotors R=+1, R=-1).

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Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).