Let $$P:[0,\pi]\times[0,2\pi)\times[0,\infty)\to\mathbb R^3\;,\;\;\;(\theta,\phi,r)\mapsto r\begin{pmatrix}\sin\theta\cos\phi\\\sin\theta\sin\phi\\\cos\theta\end{pmatrix}$$ and $r\ge0$. I would like to construct two charts$^1$ covering $$B_r:=\{x\in\mathbb R^d:\left\|x\right\|\le r\}.$$
In order to obtain a bijection, we could restrict $P$ to $U_1:=(0,\pi)\times[0,2\pi)\times[0,r]$ (so, I've excluded $\theta=0$ and $\theta=\pi$). This is clearly an open subset of $\mathbb H^3=\mathbb R^2\times[0,\infty)$. In order to ensure that the boundary of $B_r$ is mapped to $\partial\mathbb H^k=\mathbb R^2\times\{0\}$, we need to transform the last parameter. In conclusion, if $$\psi_1:U_1\to B_r\;,\;\;\;(\theta,\phi,s)\mapsto P(r-s,\theta,\phi),$$ then $\phi_1:=\psi_1^{-1}$ should be a chart for $B_r$.
Question 1: How do we see that $\Omega_1:=\psi_1(U_1)$ is an open subset of $B_r$? And is there a nice explicit form of $\phi_1$?
Question 2: The only part of $B_r$ that $\phi_1$ doesn't cover should be $$P(\{0,\pi\}\times[0,2\pi)\times[0,r])=\left\{\begin{pmatrix}0\\0\\ s\end{pmatrix}:s\in[-r,r]\right\}.\tag1$$ How can we find another chart which covers this missing part as well?
$^1$ I treat $B_r$ as a $d$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary. So, a chart $\phi$ of $B_r$ is a $C^1$-diffeomorphism from an open subset $\Omega$ of $B_r$ onto an open subset $U$ of $\mathbb H^3:=\mathbb R^2\times[0,\infty)$.