the trajectories of x, "i" = 1, ..., "n" are defined by the movement of the rigid body. this movement is given by the set of rotations ["a"("t")] and the trajectory d("t") of a reference point in the body. let the coordinates x "i" = 1, ..., "n" define these points in the moving rigid body's reference frame "m", so that the trajectories traced in the fixed frame "f" are given by