Curvature

Feb 2010
148
7
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.
 
Last edited:
Feb 2010
148
7
Sorry but 2) is wrong since \(\displaystyle \Vert \gamma''(t) \Vert \neq 1\).

Suggestions would still be appreciated.