i am encountering an issue regarding the visualization of certain sets:
For instance:
$$Σ= \{ (x, y, z) ∈ R^3 : -1\leq z\leq1, x^2+y^2 = 5(2-z^2)^2 \} $$
Could someone kindly provide an explanation on how to effectively visualize this set? Additionally, if anyone could offer recommendations for relevant videos or documentation, it would be greatly appreciated.
To understand how to visualize these sets, I employ a rather straightforward method:
I construct my Cartesian coordinate system with axes $x$, $y$, and $z$. I then rotate the paper to make the $z$-axis horizontal and proceed to graph the function that follows $x^2 + y^2 = f(z)$.
For example: $x^2 + y^2 = \sin(z)^2$ is like a little onion, with $z=[0, \pi]$, i draw the $|\sin(z)|$ in the $z$'s plain, and then I draw circles around the line.
But with something more complex than that I find it hard to draw the set.
Thank you so much for you help