How can I plot $(x^2+y^2+z^2)^2 \leq x^2+y^2$?

Question

if I use spherical coordinates $x=r\sin(a)\cos(b), y=r\sin(a)\sin(b), z=r\cos(a)$, where $a$ goes from 0 to $\pi$ and $b$ from 0 to $2\pi$ I get the following

$r^4\leq r^2\sin(a)^2$

$r^2\leq \sin^2(a)$

Do you know what shape this area has? Can I plot it with an online tool for example?

Answer 1

You are better off using cylindrical coordinates. Let $x = r\cos(\theta)$, $y = r\sin(\theta)$, and keep $z$, you have that your set is

$$ \{ (r^2 + z^2)^2 \leq r^2 \} $$

For a fixed angle, the cross sectional view of the equality case can be easily graphed by, for example, Wolfram Alpha.