Putting equations in parametric form Im told that an object has the following trajectory:
$$
\begin{gather}
T= \left\{
\begin{array}{ll}
x^2 + y^2 &= 4\\
x + z &=3
\end{array} \right.
\end{gather}
$$    
We are also told that the object makes a full tour around T.
I'm asked to put T in parametric form, but I don't understand how, since there is no $t$ in the equation.
And also, I don't understand what it means when they say that the object makes a full tour around T. Does that mean T is a circle?
 A: In the $(xy) $ plane, the trajectory is the circle of center $(0,0) $ and radius $2$.
in the space, the object moves on a cylindrical surface defined by the parametric equations:
$$x=2\cos (t) $$
$$y=2\sin (t) $$
$$z=3-x=3-2\cos (t) $$
A: Attempting an interpretation:
Let me write $\theta$  instead of  t:
$x = 2 cos(\theta)$ ;  $y = 2 sin(\theta)$ :  $z = 3 - 2 cos(\theta)$ .
In vector form:
$\vec{r}$ = $(0,0,3)$ + $2( cos(\theta), sin(\theta), - cos(\theta) )$.
$\vec{r} = (0,0,3) + \vec{r'}$
Let's look at the direction vector  $\vec{r'}$:
$\vec{r'} = 2( cos(\theta), sin(\theta), - cos(\theta) )$.
Projection of path onto the $x -, y -$ plane is a circle, I.e. the object moves on a cylinder, radius = $2$.
Cylinder axis parallel to $z$ - axis.
The $z$ component describes a $cos$ curve.
The projection of the path onto the $y-, z-$ plane  is again a circle, centre at $y = 0$, $z = 3$, I.e. the object moves on a cylinder, radius = $2$.
Cylinder axis parallel to $x$ - axis
The x component describes a $cos$ curve.
Comments welcome.
