When a ruled surface can be regular

Question

I know that a ruled surface is a surface with parametrization $x(u,v)$ = $c(u)$ + $vf(u)$ where if $I$ $\subset$ $R$ is an interval, $c$: $I$ $\mapsto$ $R^3$ and $f$ : $I$ $\mapsto$ $R^3$ are smooth curves with $f$ $\neq$ $0$ on $I$. The curve $c$ is called the directrix and the $f(u)$ are called the rulings. If I'm not mistaken, is a ruled surface regular at a point $p$ = $x(u,v)$ provided that $f(u)$ $\bigwedge$ $c'(u)$ = $vf(u)$ $\bigwedge$ $f'(u)$ and $x_u$ $\bigwedge$ $x_v$ $\neq$ $0$? I actually think that only the second condition is really required, or are both of them required? Are the two conditions equivalent, and is there any other extra requirement? I'd really appreciate some input on this, thanks.

Answer 1

Let $x(u,v)=c(u)+vf(u)$ where $u\in I$. Note that by definition, $x(u,v)$ is a regular surface if $x_u\wedge x_v\neq 0$. Simple computation gives $x_u(u,v)=c'(u)+vf'(u)$ and $x_v(u,v)=f(u)$, which implies that $$x_u\wedge x_v=c'(u)\wedge f(u)+vf'(u)\wedge f(u).$$ Therefore, $x(u,v)$ is a regular surface if and only if $c'(u)\wedge f(u)+vf'(u)\wedge f(u)\neq 0$.