Minimum radius $r=c$ central line of a Catenoid
$$ \sqrt{x^2+y^2} =c \cosh (z/c)$$
is to be mapped by isometrically bending it to a straight line (black) of length $ 2 \pi c $ without twist.
With twist, a catenoid ( right or left handed) results, as is well known and parameterized as
$$ (x,y,z)= (u \cos v, u \sin v, \pm c v); $$
Animations show $$ \text{ Right or left Helicoid } \to \text{Catenoid} \to \text{ Left or Right Helicoid } $$
However, with no twist to the black line another intermediate shape would result. For an idea of the expected symmetrical shape in the neighborhood of central line it could resemble a surface like this, call it a Frill:
So the animations could show $$ \text{ Right or left Helicoid } \to \text{Frill} \to \text{Left ot Right Helicoid } $$
and back again.
What odes or pdes in isometric mapping can define this new shape?