Translation and rotation⁠

A hinge rhomb, consisting of equal edges and using sliders that move on the fixed red rod, imple­ments axial symmetry. Indeed, the loca­tion of one of the green hinges deter­mines both the length of the oppo­site side of its triangle, and the oppo­site trian­gles are equal. That means that in any state of the mech­a­nism two green hinges remain symmetric with respect to the red rod.

Consider a figure, curvi­linear triangle, and look where it maps under the mech­a­nism action. We get a symmet­rical figure. It equals the orig­inal but is differ­ently oriented. That is, if the plane was an infi­nitely large sheet of paper with the figure drawn on it, one would have to fold it along the symmetry axis and one part would be upside-down.

Apply a symmetry mech­a­nism with a parallel axes to the triangle we obtained using the first one. The imple­ments this plane trans­for­ma­tion. So, the result of two axial symme­tries with parallel axes is a shift. The converse is also true: any parallel trans­port can be decom­posed into two symme­tries with parallel axes. It's easy to see that such a decom­po­si­tion is not unique.

Such a result of consec­u­tive mappings is called a compo­si­tion, and in terms of func­tions, a composed func­tion. As in the analyt­ical case, the result of a compo­si­tion can be obtained with consec­u­tive actions or after some simpli­fi­ca­tions. The trans­formed object may be totally different from the orig­inal.

And what happens if the symmetry axes are not parallel?

The compo­si­tion of two axial symme­tries with non-parallel axes is a rota­tion centered at the point of inter­sec­tion of the axes. And the angle of rota­tion is twice bigger than the angle between the axes. As in the case of the shift, the converse is true: every rota­tion can be decom­posed into a pair of axial symme­tries.

A linkage based on a rhomb imple­ments rota­tion trans­for­ma­tion.

What if we apply consec­u­tively a trans­la­tion and a rota­tion? Can we match the result with the source?

Decom­pose the rota­tion into two symme­tries. On the picture we can see that the stage of obtaining the gray triangle and applying a symmetry can be replaced with a single symmetry. An what is left, the compo­si­tion of two axial symme­tries with non-parallel axes, is just a rota­tion as we know.

Draw the triangle on a table. Put a sheet of paper over it and outline the figure. Lift the sheet and just a rota­tion around some point to some angle!