given is the following hyperboloid: $$H = \{(x,y,z) \mid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\},$$ where a,b,c are free parameters. I have to find an $C^\infty$-atlas for H. In order to do this, I have to find one (or more) $C^\infty$-chart, which overlays H completely. In addition to this, I should use cylindrical coordinates.
So, first of all I tried to transform H into cylindrical coordinates: $$H = \{(r,\phi,z) \mid r^2 \left(\frac{\cos(\phi)^2}{a^2} + \frac{\sin(\phi)^2}{b^2}\right) - \frac{z^2}{c^2} = 1\}$$ But now I am having heavy problems to find such a chart for this set.
I tried to do this for the special case a=b=c=1, just to get more clear about the whole thing. In this case, our Hyperboloid is $H^* = \{(r,\phi,z) \mid r^2 - z^2 = 1\}$. I think a chart for this will look like this sketch:
But I don't know how I can transform this idea to a Hyperboloid with arbitrary parameters.
I am very thankfull for any help or new idea I can get.
Best regards!
Here is a summary of the discussion in comments. The hyperboloid, being 2-dimensional, will have charts taking values in $\mathbb{R}^2$. To put it another way, each chart will have two parameters.
For the special case of the hyperboloid $r^2-z^2=1$ expressed using cyclindrical coordinates $(r,\phi,z)$, the idea is to pick two of the three coordinates $r,\phi,z$. But not any two will give a chart: picking $r$ and $z$ does not work because of the equation $r^2 = z^2+1$. Instead, the pair of coordinates $(\phi,z)$ works fine and does define charts, as long as $\phi$ has been properly restricted. Two such charts will cover the hyperboloid: one chart defined by restricting $0 < \phi < 2\pi$; and the other chart obtained by restricting $-\pi < \phi < \pi$. The actual functions that define these charts are obtained from the standard formulas expressing cylindrical coordinates in terms of Euclidean coordinates.