 Solution

For fixed $u$, varying $v$ traces a unit circle in the radial-vertical plane centred atThus ${\Sigma }_1$ is a unit tube whose centre winds helically around the $z$-axis. Sincethe given curve is its inner helix.

Take ${\Sigma }_2$ to be the circular cylinder $x^2+y^2=1$. Along $v=\pi$,so their span is the tangent plane of that cylinder. The two surfaces are therefore tangent along $\gamma$. Moreover,which is normal to the cylinder, so $\gamma$ is a geodesic of ${\Sigma }_2$. Part (ii) now proves that it is a geodesic of ${\Sigma }_1$. This is the cylindrical-helix tangency construction.