I will first introduce my notations then ask my questions. Thank you in advance for your answer.
Notations:
Given a surface of revolution $S_\Gamma$ of profile curve $\Gamma$ of class $C^k$ given by
$$\begin{equation} \Gamma = (r, z) : [a,b] \rightarrow \mathbb{R}^2 : t \mapsto (r(t), z(t)), \end{equation}$$ where $r(a)=r(b)=0, r|_{]a,b[}>0 $, one can describe $S_\Gamma$ (excepting the poles) with the local parametrization
$$\begin{equation} X : ]a,b[ \times \mathbb{R} \rightarrow \mathbb{R}^3 : (t,\theta) \mapsto (r(t)\cos(\theta),r(t)\sin(\theta), z(t)). \end{equation}$$
We consider the open set of $S$ containing the poles $$U_N = \{N\} \cup \{X(s,\theta) | s < a + \varepsilon\}, \; \; U_S = \{S\} \cup \{X(s,\theta)| s > b - \varepsilon\} $$ and the following associated charts $$\phi_N : U_N \rightarrow \mathbb{R}^2 : \begin{cases} N \mapsto (0,0),\\ X(s, \theta) \rightarrow ((s-a) \cos (\theta) ,(s-a) \sin(\theta)), \end{cases} \\ \phi_S : U_S \rightarrow \mathbb{R}^2 : \begin{cases} S \mapsto (0,0),\\ X(s, \theta) \rightarrow ((b-s) \cos (\theta) ,(b-s) \sin(\theta)). \end{cases} $$
Questions:
I have two questions that are strongly related :
-
- Is $$\mathcal{A}=\{(S_\Gamma\setminus (\{N,S\} \cup X(]a,b[,0)), X|_{]a,b[\times ]0,2\pi[}^{-1}),(S_\Gamma\setminus (\{N,S\} \cup X(]a,b[,1)), X|_{]a,b[\times ]1,2\pi +1[}^{-1}),\\ (U_N, \phi_N),\\ (U_S, \phi_S)\}$$ an atlas of $S_\Gamma$ ? If it's not, can you give me another atlas ? Maybe my chart $\phi_N, \phi_S$ are not good enough...
-
- What are the sufficient conditions on the profile curve $\Gamma = (r,z)$ in order to have $S_\Gamma$ a surface of class $C^k$ ?
I believe the answer is about the behavior and the regularities of the derivative of $r$ and $z$ at the limits $a$ and $b$, but I can't express it.
Thank you for your answer !