Ruled surfaces

A ruled surface is a surface constructed moving a straight-line along a given curve. If $\alpha=\alpha(s)$ is this curve and $w(s)$ is the direction of the straight-line at $\alpha(s)$, the straight-line is the set $\{\alpha(s)+t w(s):t\in{\mathbb R}\}$. Thus the surface $S=\{\alpha(s)+t w(s): s\in I,t\in{\mathbb R}\}$ and the parametrization is $X(s,t)=\alpha(s)+t w(s)$. Since $X_s=\alpha'(s)+tw'(s)$ and $X_t=w(s)$, then we have to assume that they are independent linearly. Then $X$ is a parametrized surface and so, locally, $S$ is a surface.   

We show some examples. Suppose $\alpha$ is a planar curve included in the plane $z=0$. If we take $w(s)=a=(0,0,1)$, we have the right cylinder over the curve $\alpha$. In the next figure, $\alpha$ is the parabola $\alpha(s)=(s,s^2,0)$ and $w(s)=(0,0,1)$.

We can take $w(s)$ to be tilted at each point of $s$. If $\{T(s), N(s), B(s)\}$ is the Frenet trihedron along $\alpha$,  then up to reparametrizations, $B(s)=(0,0,1)$. If we take $w(s)=\cos(m) N(s)+\sin(m) B(s)$, with $m\in{\mathbb R}$,  we obtain a cone along $\alpha$. In the next picture,  $\alpha$ is the parabola again.

If we replace the constant $m$ by a function $\theta(s)$, then $w(s)$ goes changing at each point. Here we take $\alpha$ the circle $\alpha(s)=(\cos(s),\sin(s),0)$. If $w(s)$ is a $2\pi$-periodic function, then $w$ is also $2\pi$-periodic. This occurs for example if $\theta(s)=s$. The parametrization is $X(s,t)=(\cos (s)-t \cos ^2(s),\sin (s)-t \sin (s) \cos (s),t \sin (s))$ and the surface is:

But if $w$ is $4\pi$-periodic, then we obtain a Möbius strip. For this, we take $\theta(s)=s/2$. Then $$X(s,t)=\left(\cos (s) \left(1-t \cos \left(\frac{s}{2}\right)\right),\sin (s) \left(1-t \cos \left(\frac{s}{2}\right)\right),t \sin \left(\frac{s}{2}\right)\right).$$