If Cartesian coordinates have an $x$-axis, $y$-axis, and $z$-axis, do spherical coordinates have an $r$-axis, a $\theta$-axis, and a $\phi$-axis?
Since the Cartesian $z$-axis is just the set:
$$\{(0, 0, z) \mid z \in \mathbb{R}\}$$
I assume that the $r$-axis would be the set:
$$\{(r, 0, 0) \mid r \in \mathbb{R} \}$$
although I guess we have to stipulate $r \ge 0$. How to picture this? Since $\phi = 0$, we'd be pointing straight up, so that leads me to think that the $r$-axis in spherical coordinates is (half) the $z$-axis in Cartesian coordinates. Is this correct?
What about the $\theta$-axis and the $\phi$-axis? I want to picture the first as a circle in the $xy$-plane, but since $r = 0$ maybe it would just all collapse to zero. Likewise for $\phi$ -- I want to see it as an arc extending from the North Pole to the South Pole, but again $r = 0$.
Does "the set of all points where all but coordinate $c$ are zero" suffice as a description of $c$-axis? Are there coordinate systems with axes that aren't linear subspaces? (I thought spherical or polar would do this, but I seem to have failed!)