Computing E, F, G, L, M, N and asymptotic/curvature-line nets

A surface $S \subset \mathbb {R}^{3}$ is called triply ruled if at every $p \in S$ we can find three open line segments $L_1, L_2, L_3$ lying in $S$ such that $L_1 \cap L_2 \cap L_3 = \{ p\}$. Determine all triply ruled surfaces.

Consider the torus-of-revolution $T$ obtained by rotating the circle $(x-a)^2+z^2=r^2$ around the $z$-axis:

Parameterize this torus, compute its Gauss curvature function $K$, and verify that $\int_T K \, dA = 0$ by explicit calculation.

Let $\alpha : (0,1) \to \mathbb {R}^{3}$ be any regular arc (that is, $\alpha$ is differentiable and $\alpha '$ is nowhere zero). Let $T(u), N(u)$ and $B(u)$ be the unit tangent, normal and binormal vectors to $\alpha$ at $\alpha (u)$. Consider the normal tube of radius $\varepsilon$ around $\alpha$, that is, the surface given parametrically by $\phi (u,v) = \alpha (u) + \varepsilon \cos (v) N(u) + \varepsilon \sin (v) B(u)$.

Let $(f(v)\cos (u), f(v)\sin (u), g(v))$ be a parametrization of a surface of revolution $S \subset \mathbb {R}^{3}$ where $(u,v) \in (0,2\pi )\times (a,b)$. If $S$ is given the induced metric from $\mathbb {R}^{3}$, prove that the following map from $S$ to $\mathbb {R}^{2}$ is locally conformal where $\mathbb {R}^{2}$ is given the standard Euclidean metric: