= Solution
Using the two <first fundamental forms> calculated below, choose
$$
\boxed{f(s)=\cosh s.}
$$
The <surface of revolution> is then a <catenoid>, with metric $\cosh^2s\,(ds^2+dv^2)$. On the <helicoid>, take $u=\sinh s/a$, assuming $a\ne0$ so that its parametrization is regular. Substitution gives
$$
a^2du^2+(1+a^2u^2)dv^2
=\cosh^2s\,ds^2+\cosh^2s\,dv^2.
$$
Thus the coordinate map
$$
\left(\sinh s\cos v,\sinh s\sin v,v\right)
\longmapsto
\left(\cosh s\cos v,\cosh s\sin v,s\right)
$$
is a <local isometry> on sufficiently small patches. The constant $a$ only changes the radial parametrization of the given <helicoid>, whose pitch is fixed at one. For $a=0$ the displayed <helicoid> parametrization is not a regular <surface>.
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