I am trying to construct a parametrization of a surface of revolution. By parametrization, I mean a patch for the surface that covers the entire surface. I am having trouble with even starting to find a parametrization. My question is how does one go about constructing a parametrization for a surface? For example:
(i) The surface of revolution of $y=cosh x$ around the x axis is a catenoid.
(ii) The surface of revolution of the circle $(x-2)^2+y^2=1$ around the y axis is a torus.
I'd like to do these myself so I request some guidance. Thanks for your time.
You know $y =coshx = \dfrac{e^x+e^{-x}}{2}$
So the parameterization would be $x = t$ and $y = \dfrac{e^t+e^{-t}}{2}$
I think
For the next one it is straight forward
$x = 2+cost$ and $y = sint$