Given a family of lines $(\gamma_t): a(t)x+b(t)y+c(t)=0,\ t\in I$ (interval in $\mathbb{R}$), where $ a,b,c:I\to\mathbb{R},\ a^2(t)+b^2(t)\neq 0,\ \forall t\in I$ , what conditions should we impose over $a,b,c$ such that we can be sure that this family admits a unique envelope?
What is the most general case?
I found it good if $a,b,c\in C^1 (I),\ a'(t)b(t)-a(t)b'(t)\neq 0,\ \forall t\in I$. Can it be done more than that?
What if the function $t\mapsto a'(t)b(t)-a(t)b'(t),\ t\in I$ has a finite number of zeros? (or an isolated set of zeros)?
If you have some references I'll be thankful, because I didn't find too much on that problem.
