Find a parameterization of the paraboloid $900z = 25x^2 + 36y^2$, $z \le 16$.
The solution is $r(u, v) = (6v\cos(u), 5v\sin(u), v^2)$ where $0 \le u \le 2\pi$, $0 \le v \le 4$.
This is the first time I've been asked to parameterise a paraboloid, so I'm struggling to deduce the reasoning behind this solution. I can tell the solution utilises cylindrical coordinates, but it's done so in a way that I've never seen.
I would greatly appreciate it if people could please take the time to show the calculations and reasoning behind this solution. This will help me understand the solution and, therefore, solve similar paraboloid parameterisation problems in the future.
I recently asked a question relating to a similar problem from the same problem set. These are two different problems, and the questions relating to them are also completely different. I would like to thank Bubba for his assistance with the previous question.
Best Answer
Let's divide your equation by $900=25\cdot 36$ You get $$z=\frac{x^2}{6^2}+\frac{y^2}{5^2}$$ If we do the transformation $x/6=X$, and $y/5=Y$, this looks very much like the equation of a circle $X^2+Y^2=R^2$. In this case $R^2=z$. The parametric equation of the circle is then $X=R\cos\theta$ and $Y=R\sin\theta$. So you have $x=6R\cos\theta$, $y=5R\sin\theta$, $z=R^2$, where $\theta$ varies between $0$ and $2\pi$. Since $z$ is between $0$ and $16$, it means $R$ is between $0$ and $4$. If you rename your variables from $R$ to $v$, and from $\theta$ to $u$, you get your result