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

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 $$\mu(s)$$?

• If a particle travels (with undetermined speed) along a circle centered at the origin, do you conclude that its velocity must be $0$? The unit tangent vector cannot be the $0$ vector, by the way. Go back to basics. Commented Jan 7, 2023 at 22:25
• @TedShifrin thanks for the picture in my head. Now it is clear to me. Can I make the assumtion in my proof, that $\beta(s)= \left(Rcos\left(\frac{s}{R}\right),Rsin\left(\frac{s}{R}\right)\right)$ the only curve which satisfies these properties, and use this?
– T_B
Commented Jan 8, 2023 at 15:04
• No, because we’re talking about space curves, not planar curves. Commented Jan 8, 2023 at 15:52

For part (a) consider taking the derivative of $$\|\beta(s)\|^2 = k$$ for some constant $$k$$.
• I'm taking the liberty of replacing your $\beta(s)^2$ with $\|\beta(s)\|^2$. Although some people write $v^2$ for $\|v\|^2$, it is mostly just very confusing. Do you also write $vw$ for $v\cdot w$? Commented Jan 8, 2023 at 0:43