A geometry problem regarding a surface of revolution ... Prove that surface $x(u_1,u_2)=(a(\cos u_1+\cos u_2), a(\sin u_1+\sin u_2), b(u_1+u_2))$  has an isometric correspondence with a revolution surface.
 A: Hint: Changing variables by, $\psi=\frac{u_1-u_2}{2}$, $\phi=\frac{u_1+u_2}{2}$,
$$a(\cos u_1+\cos u_2)=2a\cos{\frac{u_1-u_2}{2}}\cos{\frac{u_1+u_2}{2}}=2a\cos{\psi}\cos{\phi}\\
a(\sin u_1+\sin u_2)=2a\cos{\frac{u_1-u_2}{2}}\sin{\frac{u_1+u_2}{2}}=2a\cos{\psi}\sin{\phi}\\
b(u_1+u_2)=2b\phi.$$
This very closely resembles the parametric form of a surface of revolution about the $z$-axis, but we're not quite there. Can you think of a second change of variables that'll finish the job?
A: Yes, one has to find a mapping which makes the metrics identical. In the case of your helicoidal surface the surface of revolution should be a catenoid in some form. See: http://en.wikipedia.org/wiki/Catenoid. 
A: You can consult:
1- do Carmo, M. P., "Helicoidal surfaces with constant mean curvature" Tôhoku Math. J. 34, 425-435 (1982). 
2- L. R. Hitt and I. Roussos "Computer graphics of helicoidal surfaces with constant mean curvature", Anais da Academia Brasileira de Ciências 63 (3), 211 (1991).
(https://www.dropbox.com/s/6yktpcxpiqavdcs/AnAcadBrasCi.63.211_1991_ComputerGraphicCMCsurfaces.pdf?dl=0)
