Formulas for space curves

Question

We had the following theorem and I have two questions about it:

Definition A curve $c:I \to \mathbb{R}^3$ is said strictly regular, if $c'(t)$ and $c''(t)$ are linearly independent.

Theorem Let $c$ be a strictly regular curve. Then $$T=\frac{c'(t)}{\|c'(t)\|}\qquad B=\frac{c'\times c''}{\|c'\times c''\|}\qquad N=B\times T$$ $$\kappa(t)=\frac{\|c'\times c''\|}{\|c'\|^3}\qquad\tau(t)=\frac{\langle c'\times c'',c'''\rangle }{\|c'\times c''\|^2}$$

1. Is it true that if $c$ is strictly regular $\implies$ $c$ is a regular curve? My argument would be because if $c'(t)=0$ then $c''(t)$ is always linearly dependent to the zero-vector.

2. Does the theorem also hold for just regular curves or even just curves? If not, which formulas do still hold?

Answer 1

The answer to your first question is yes, of course. Any set of vectors with the $0$-vector among them is linearly dependent.

Assuming you have a regular curve, there will be no principal normal (hence no $N$ or $B$) defined at points where $\kappa=0$. These are precisely the points where $c',c''$ become linearly dependent. So the whole game comes to a grinding halt. You still are fine with the formula for $T$ and we do get $\kappa=0$. But then we can go no farther.