
Curvature
Hello all,
I have a regular parametrized curve $\displaystyle \gamma: \mathbb{R} \rightarrow \mathbb{R}^{3}$ such that $\displaystyle \Vert\gamma''(t)\Vert =1 $ for all $\displaystyle t \in \mathbb{R}$.
Assume that $\displaystyle \gamma(t)$ has constant curvature $\displaystyle k \neq 0$ and constant torsio $\displaystyle \tau=\frac{1}{\sqrt{2}}$. We also assume that:
$\displaystyle \gamma(0)=\left( \frac{1}{\sqrt{2}},0,0 \right)$
$\displaystyle \gamma'(0)=\left( 0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)$
$\displaystyle \mathbf{b}(t)=\frac{1}{\sqrt{2}} \left( \sin(t),\cos(t),1 \right)$
where $\displaystyle \mathbf{b}(t)$ is the binormal of $\displaystyle \gamma(t)$
I am asked to:
1) find the curvature K
2) find the $\displaystyle \gamma(t)$ explicitely
I have done the following:
1)
Using the Frenet equations I have that $\displaystyle \mathbf{n'}=k\mathbf{t}+\tau\mathbf{b}$.
Differentiating on both sides yields $\displaystyle \mathbf{n''}=k\mathbf{t'}+\tau\mathbf{b'}$ (*).
Using $\displaystyle \mathbf{b'}=\tau\mathbf{n}$ and $\displaystyle \mathbf{t'}=k\mathbf{n}$ and substituting in (*) gives $\displaystyle k=\frac{\sqrt{2}}{2}$.
2)
I am still not done with this one but my suggestion is to use that $\displaystyle \mathbf{n}=\frac{\gamma''(t)}{\Vert \gamma''(t) \Vert}$ and integrate together with the fact that $\displaystyle \Vert\gamma''(t)t\Vert =1 $.
Could someone verify 1) and 2)?
Thanks.

Re: Curvature
Sorry but 2) is wrong since $\displaystyle \Vert \gamma''(t) \Vert \neq 1$.
Suggestions would still be appreciated.