There are at least three special kinds of rigid transformations. These are, not in any order of importance, translation (T), rotation (R), and reflection (M). Using Cartesian coordinate geometry of 2D Euclidean spacetime, the following mappings always hold true.

[math]T(x,t)=(x+\\Delta x, t+ \\Delta t)[/math]

[math]R(x,t)=(xcos\theta-tsin\theta, xsin\theta+tcos\theta)[/math]

[math]M_S(x,t)=(x, -t)[/math] reflection in the space axis.

[math]M_T(x,t)=(-x, t)[/math] reflection in the time axis.

The distinction of all rigid transformations is that they do not stretch (dilate), shrink (contract), or twist (curve) the transformed events. Two transformed events do not move closer together or farther apart in spacetime. In other words, events always stay the same distance from each other. Furthermore, the Pythagorean Theorem determines this distance. The mechanics is Newtonian and the spacetime is Galilean.

Nonrigid transformations give relativistic mechanics and Minkowskian spacetime. For transformed events involving dilation and contraction refer to Einstein’s special theory of relativity and for the ones involving twisting and curvatures refer to Einstein’s general theory of relativity.