The accompanying figure shows the graphs of two parametric representations of the cone $z=\sqrt{x^2+y^2}$ for $0\leq z\leq2.$
$(a)$ Find parametric equations that produce reasonable facsimiles of these surfaces.
For ${\rm I}$, I notice that in $xy-$plane it's just circle which vary for $z$. Using is fact, $$x=r\cos\theta,\quad x=r\sin\theta,\quad z=r^2 \text{ where } 0\leq r\leq \sqrt 2,0\leq \theta\leq 2\pi$$ But how to solve for ${\rm II}$. Please answer it as I can also figure it out for general surfaces also.
Thanks in advance and thanks for your time .