As a consequence of Gauss's Theorema Egreguim, a nessecary condition for two surfaces to be isometric to each other is that the Gaussian curvature of corresponding points should be the same. However, this condition is not sufficient, and I read that there are surfaces fulfilling this criterion that are not globally isomertic to each other. I'm willing to learn a bit more about the sufficient conditions of applicability of two surfaces to each other, as well as the different approaches to finding an explicit transformation of a given surface into another surface.
As an example, Wikipedia article Catenoid mentions such an explicit formula, for the transformation of the catenoid into an helicoid. First of all, the parametric representation of a catenoid whose axis of symmetry is $z$ is:
$$z=v$$ $$x=c\cdot \mathbb{cosh}(v/c) \mathbb{cos} u$$ $$y=c\cdot \mathbb{cosh}(v/c) \mathbb{sin} u$$
and by using the following transformation (called "helicoid transformation" in the wikipedia article):
$$x(u,v)=\mathbb{cos\theta\cdot sinhv\cdot sin} u + \mathbb{sin\theta\cdot coshv\cdot cos} u$$ $$y(u,v)=\mathbb{-cos\theta\cdot sinh v\cdot cos} u +\mathbb{sin\theta\cdot coshv\cdot sin} u$$ $$z(u,v)=u\mathbb{cos}\theta+v\mathbb{sin}\theta$$
one gets a family of surfaces isometric to the initial catenoid (with the continous parameter for this family of surfaces being $\theta$); an helicoid corresponds to the value $\theta =\frac{\pi} {2}$.
So my question is really about understanding more on the principles behind such mappings. Good references that discuss this particular problem will also be blessed!