An undefined Frill form shown here ( schematically in appearance similar to $ z = \cos x \cosh y $ but is not its exact isometric form ).. is isometric to the catenoid and helicoid.
The Frill is symmetric about $(xz, yz) $ planes seen with 4 waves, the helicoid 2 helical turns and catenoid with repeated scrolls. $ z= f(x,y) $ in Monge form like the one pictured does not have $ K=-1$
The catenoid and helicoid inter-deformation consists of bending/twisting. The mapping and parametrization are known.
Please help find
parametrization of the Frill
and so its $H,$ with $K$ invariance.
and its mapping separately to arrive at the catenoid and helicoid... by pure bending and pure twisting parametrizations starting at the Frill.
K and H of $ z = \cos x \cosh y $ for comparison.
EDIT1:
It is required to find if the isometric variant of the Frill with same first fundamental form metric is a minimal surface $H=0 $ along with the catenoid and helicoid.
