Spherical Coordinates — Question 8

Question 8 – Solution

Strategy. On the axis, the horizontal projection has length $\rho\sin\phi=0$, so analyze the poles and origin separately.

Step 1: Positive axis If $a>0$, then $\rho=a$, $\phi=0$, and $\theta$ is arbitrary: $$\boxed{(a,\theta,0),\quad 0\le\theta<2\pi}.$$

Step 2: Negative axis If $a<0$, then $\rho=|a|$, $\phi=\pi$, and again $\theta$ is arbitrary: $$\boxed{(|a|,\theta,\pi),\quad 0\le\theta<2\pi}.$$ At the origin, $\rho=0$ and both angles are arbitrary in their stated ranges.

Step 3: Consistency At $\phi=0$ or $\pi$, $\sin\phi=0$, so $x=y=0$ regardless of $\theta$, while $z=\rho\cos\phi$ has the correct sign. At $\rho=0$, all three Cartesian coordinates vanish regardless of either angle. Thus the nonuniqueness is coordinate degeneracy, not geometric ambiguity.