# Failures of Spherical Coordinates: What do we do with points on the $z$-axis?

For a set of spherical coordinates ($$\rho, \theta, \phi$$), we'll define $$\rho= \sqrt{x^2 + y^2 + z^2}$$, $$\theta \in [0, 2\pi]$$ as the rotation away from the $$x$$-axis (forward-back horizontal axis), and $$\phi \in [0, \pi]$$ as the rotation down from the $$z$$-axis (vertical axis). So, routine stuff, but just to dispel any ambiguities.

Consider the point with Cartesian/ rectangular coordinates (0,0,10); really we could generalize this to some sort of (0,0,$$z$$) for $$z \in \mathbb{R}$$. Suppose we want to convert this into spherical coordinates.

What are we supposed to do here?

From some self-teaching, I've essentially just thought of going from rectangular to spherical coordinates as solving this "system":

$$\begin{cases} x = \rho \sin\phi \cos \theta \\ y = \rho \sin \phi \sin \theta \\ z = \rho \cos \phi \end{cases}$$

It's quickly clear that we run into an issue, and I haven't yet found a definitive answer anywhere; only tangentially related answers and wiki's. $$\phi = 0 \implies \sin \phi = 0$$, so we cannot determine what $$\theta$$ is.

Thinking geometrically, wouldn't we be able to give any theta considering we're going straight up the $$z$$-axis? Would that not indicate the same point? Testing my hypotheses with some online calculators yielded nothing: the ones I tried just didn't spit out an answer, so the other possibility I can imagine is just that there is... well, no spherical representation for these such points. I assume this system is rigorous, well-defined, etc. (and that, therefore, there is an answer to this), but this seems to be quite an oversight.

TLDR; how do we handle points on the $$z$$-axis with the spherical coordinate system?

• Even when we take $\theta \in [0, 2\pi)$ we are faced with a violation of one-to-one correspondence: the plane $r = 0$ of the space $r\phi \theta$ is mapped to the origin of the coordinates $x = y = z = 0$, a line $\phi=0(\pi), r=r$ is mapped to one point: $x=y=0,z=r$. Jan 24, 2021 at 17:22