$\beta:I\subset\mathbb{R}\to\mathbb{E}^3$ an arc length parameterized curve, how can I prove following statements:

Question

We know that: $$\beta:I\subset\mathbb{R}\to\mathbb{E}^3$$ is an arc length parameterized curve with $$\forall s\in I:\kappa(s)\neq0 \wedge \tau(s)\neq0$$ And knowing that: $$\Vert\beta(s)\Vert=const.$$

a) how can I prove that $\beta(s)\bullet\beta’(s)=0\space(\forall s\in I)$

So far I came up with, that $\Vert\beta(s)\Vert=const. \implies \beta’(s)=0$ is this correct or am I here mistaken?

b)prove that there exist functions $\lambda(s),\mu(s):\beta(s)=(\lambda(s)\bullet N(s)+\mu(s)\bullet B(s)$

*how could I best do this? I thought so far, N and B are linearly independent, and form a positive basis, and because $T_{\beta}=\beta’=0\space \forall s \in I$ We can describe $\beta$ entirely by N and B, but how do I get to $\lambda(s)$ and $ \mu(s)$?

Thank for any help already in advance.

Answer 1

Hints:

For part (a) consider taking the derivative of $\|\beta(s)\|^2 = k$ for some constant $k$.

For part (b), take a look at the Frenet-Serret formulas.