I found the following:
Cylindrical coordinates $(\rho , \theta , z)$.
This system consists of the following coordinate surfaces:
- Cylinders with common $z-$axis: $\rho=\sqrt{x^2+y^2}=\text{ constant }$
- Semiplanes that pass through the $z-$axis: $\theta=\arctan \left (\frac{y}{x}\right )=\text{ constant } $
- Planes parallel to the plane $xy$: $z=\text{ constant }$.
$$$$
Could you explain to me why $\rho=\sqrt{x^2+y^2}=\text{ constant }$ means "cylinders with common $z-$axis", $\theta=\arctan \left (\frac{y}{x}\right )=\text{ constant } $ means "semiplanes that passes through the $z-$axis" and $z=\text{ constant }$ means "planes parallel to the plane $xy$" ??
if you set $\sqrt{x^2 + y^2} = \rho$, you get a circle of radius $\rho$ on every plane $z = constant.$ what kind of object produces a circle of constant radius when cut by parallel planes?
the reason you get semi planes instead of planes is you set $\theta = \tan^{-1}(y/x) = constant $ then $-\pi/2 < \theta < \pi/2$ and you have set $\rho > 0$ the points $\rho, \theta$ won't be able to reach the second or third quadrant.