The group 2C_1 has 2 rotations x->x'= R~xR (aa=1, bb=1). One with "positive" sense represented by the rotor
R = 1 = (ab)^0
and one corresponding rotation with "negative" sense represented by the rotor
R = ba = -ab = -1.
Here the rotations are realized by double reflections x->x' -- similar to the applet Rotations by reflection. The final result x' after the second reflection (at the red line of reflection) is shown as another blue ring marked by "1'". One can "see" the above 2 rotations by interactively changing the position of the bright red point (this changes the second red line of reflection). We only get a symmetry when the red vector coincides with the [positive] blue vector a, or with the [negative] green vector b. These 2 symmetry operations are in 1:1 correspondence with the 2 rotors listed above. Combining the 2 single reflections, we get the (identity) rotations x->x'=R~xR (=x) of the symmetry group of a non-symmetric figure (R=+1, R=-1).
Remark: What "puzzles" me is that in the above figure a reflection about the a-line leaves the figure invariant. It seems therefore that the object is not completely void of symmetry. If you have any idea about this, please tell me: Eckhard Hitzer
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).