how do we decide if a trajectory is 1D, 2D or 3D?

Hello Forum,

In kinematics we study motion and the trajectories of moving bodies.

The trajectory is a line (straight or curved) that joins all the positions occupied by the object in the various instants of time. A trajectory has an equation that contains only spatial coordinates (not time t).

For example, a particle moving in a circle in the 3D space: the trajectory equation can be x^2+y^2=16 in Cartesian, r=4 in polar, etc....

Is this trajectory and this motion 2D, 1D or 3D? How do we decide?

There seem to be only one independent variable in x^2+y^2=16 ......

Is a curve always a 1-dimensional object, manifold that lives in a higher dimension space?

thanks
fisico30