When a particle moves in the space, during a time-interval *I*, we think of its coordinates as functions of the variable *t* (= time), namely
(*x*=*x*(*t*),*y*=*y*(*t*),*z*=*z*(*t*)). The set of all the locations of the particle is a *curve*
in the space, named *the trajectory* of the particule, and the functions *x*,*y*,*z* define a *parametrization* of
.

The vector
is the *position vector* of the particle at time *t*, and the point *M* such that
is the *position* of the particle at time *t*.

Without the *z*-component, we have the corresponding definitions for a curve/ the motion of a particule in the plane.

We have:
,
i.e. the trajectory of the particle is on the parabola whose equation is *y*=1-2*x*^{2}. Is the trajectory the whole parabola? No! because
.
With the notations of Figure 2 the trajectory is the arc *AC*, on which the particule travels alternatively in both directions, with a period of .

The trajectory is the unit circle in the plane whose equation is

The trajectory is a circular helix, whose projection onto the