So, we know that any circle centered at the origin $(0,0)$ can be written in polar coordinates as:
$r=f(\theta)=C \iff \langle\theta(t), r(t)\rangle=\langle t,C\rangle$
Where $C\in \mathbb{R}$ is the radius of the circle.
However, let's say I move my circle to any point in the polar plane, if this was cartesian coordinates, one can simpy state that the original circle was parametrized as:
$\langle x,y\rangle=\langle C\cos(t),C\sin(t)\rangle$
So shifting it is simply adding an offset vector, giving us:
$\langle x,y\rangle=\langle C\cos(t)+v_x, C\sin(t)+v_y\rangle$
Is it correct to say that the parametric function:
$\langle \theta,r\rangle=\langle t+v_\theta,C+v_r\rangle$ is moving the circle $v_\theta$ radians in the counter-clockwise direction and $v_r$ length units away from the origin?