Catenoid-Helicoid isometry

The parametrization

\[ \begin{array}{l} x(u,v):=(\cosh u\cos v\cos\alpha+\sinh u\sin v\sin\alpha), \\ y(u,v):=(\cosh u\sin v\cos\alpha-\sinh u\cos v\sin\alpha), \\ z(u,v):=(u\cos\alpha+v\sin\alpha) \end{array} \]

represents a one-parameter family of conformally parametrized minimal surfaces; \( \alpha=0 \) gives the catenoid, \( \alpha=\frac{\pi}{2} \) the helicoid

generated with ps-tricks

Peckham, S.D.: Solutions to nonlinear partial differential equations: a geometric approach (1999)