Parametrizing the surface $z=7-x^2-4y^2$

Question

I am willing to parameterize the surface formed by paraboloid $z=7-x^2-4y^2$ bounded below by the plane $z=3$. i know its simple that parameterize is $x=u$, $y=v$ and $z=7-u^2-4v^2$ But i am unable to decide range of values of $u$ and $v$

Answer 1

The intersection of the plane $z=3\;$ and the parabolic surface is the ellipse $$\frac{x^2}{4}+{y^2}=1$$ which can be parameterized by generalized polar coordinates $(r,\varphi)$ as $$\begin{aligned}x=&\; 2r\cos\varphi \\y=&\; r\sin\varphi \\z=&\; 7-4r^2\end{aligned}$$ Here $\varphi\in[0,2\pi),\;r\in[0,1].$