= Local isometry between the helicoid and catenoid
The <helicoid> $H(s,v)=(\sinh s\cos v,\sinh s\sin v,v)$ and <catenoid> $C(s,v)=(\cosh s\cos v,\cosh s\sin v,s)$ have the same <first fundamental form> $\cosh^2s\,(ds^2+dv^2)$. Therefore matching equal parameter pairs gives a <local isometry>. The result is local: a full turn changes the height on the <helicoid> while the angular coordinate on the <catenoid> repeats. Equal intrinsic lengths do not require the same embedding into Euclidean space.
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