How could we certify that two figures on the plane have the same shape? An approach that seems to be simple is to move one figure to place it on top of the other and, once this is done, check that they match, i.e. that their overlapping is perfect. In a bricolage implementation of this idea, one of the figures would be drawn on a fixed, say white, board and the other on a transparency. In this implementation, “to move one figure” is equivalent to moving the transparency.
If the transparency is initially placed on top of the white board, to each point of the former there corresponds a point on the board, namely the point “below”. After moving it to attempt a matching, each point on the transparency corresponds to another point on the board. This correspondence induces, in turn, a correspondence between points on the board. Indeed, for every point Pb on the board we consider the point Pt above it on the transparency. Then we move the transparency and consider the new point P′b which is now below Pt. A way (admittedly convoluted looking at first) to describe the movement itself would consist of describing the transformation that associates to each point Pb on the board the corresponding P′b. This chapter pursues this idea. In doing so, we will eventually prove that there are, essentially, just a few ways to move figures on the plane.
Arguably the simplest of the transformations on the plane, a translation maps every point on the plane to the only point at a given distance in a given direction.
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