(i) The catenoid is the surface $C$ in Euclidean $\mathbb{R}^{3}$, with co-ordinates $x, y, z$ and Riemannian metric $d s^{2}=d x^{2}+d y^{2}+d z^{2}$ obtained by rotating the curve $y=\cosh x$ about the $x$-axis, while the helicoid is the surface $H$ swept out by a line which lies along the $x$-axis at time $t=0$, and at time $t=t_{0}$ is perpendicular to the $z$-axis, passes through the point $\left(0,0, t_{0}\right)$ and makes an angle $t_{0}$ with the $x$-axis.

Find co-ordinates on each of $C$ and $H$ and write $x, y, z$ in terms of these co-ordinates.

(ii) Compute the induced Riemannian metrics on $C$ and $H$ in terms of suitable coordinates. Show that $H$ and $C$ are locally isometric. By considering the $x$-axis in $H$, show that this local isometry cannot be extended to a rigid motion of any open subset of Euclidean $\mathbb{R}^{3}$.