What is T, N, and B in terms of vectors and curvature?

Question

Not exactly sure what they want here?

Is T the derivative and N is the normal vector? What's B then? There is no prior reference to these letters in my textbook my teacher's notes/slides, or in previous questions. Any ideas are welcome.

Answer 1

$T,N$ and $B$ are the unit tangent, unit normal and binormal vectors respectively.

Provided that $r'(t)\neq 0 $ you have $$T(t)=\frac{r'(t)}{||r'(t)||}$$

$$N(t)=\frac{T'(t)}{||T'(t)||}$$ and $$B(t)=T(t) \times N(t).$$

Also see the Frenet–Serret formulas and here.

Answer 2

$B$ is $T\times N$, the cross product. There has to be some reference somewhere. This is standard differential geometry. $T,N$ and $B$ constitute what's known as a moving frame. You can look up Frenet-Serret formulas.

Answer 3

These are the Frenet-Serret formulas for curves in $\mathbb R^3$. $B$ is the binormal vector. $T$ is tangent vector and $N$ the normal vector.

https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas