Here is a simpler version of my question. Given curve $\displaystyle y=f(x)$ rotated around the y-axis. The generated surface can be described with parameters $\displaystyle x$ and $\displaystyle \theta$. Which parametrization below is correct for that surface?
A) $\displaystyle (f(x)\cos(\theta ),f(x),f(x)\sin(\theta ))$
or
B) $\displaystyle (x\cos(\theta ),f(x),x\sin(\theta ))$
I think B is correct, and I think A is incorrect. But, I usually trust Wikipedia and it essentially states that A is the correct answer. See section Coordinate expressions on Wikipedia page