kidsmoker
Jul15-09, 07:22 AM
1. The problem statement, all variables and given/known data
Let \underline{r} be a regular parameterisation of a space curve C \subset R^{3}. Prove that
\kappa=\frac{\left\|\underline{\dot{r}}\times\unde rline{\ddot{r}}\right\|}{\left\|\underline{\dot{r} }\right\|^{3}} .
3. The attempt at a solution
We have
t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\rig ht\|}
so differentiating both sides wrt u we obtain
\frac{dt}{du}=\frac{\frac{d^{2}r}{du^{2}}}{\left\| \frac{dr}{du}\right\|}+\frac{dr}{du}\frac{d}{du}(\ frac{1}{\left\|\frac{dr}{du}\right\|}).
Since
\frac{dt}{du}=\kappa\underline{n}
this gets me the curavture in terms of the desired bits (with n too) but I can't seem to get it to the desired result :\
Thanks for your help!
Let \underline{r} be a regular parameterisation of a space curve C \subset R^{3}. Prove that
\kappa=\frac{\left\|\underline{\dot{r}}\times\unde rline{\ddot{r}}\right\|}{\left\|\underline{\dot{r} }\right\|^{3}} .
3. The attempt at a solution
We have
t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\rig ht\|}
so differentiating both sides wrt u we obtain
\frac{dt}{du}=\frac{\frac{d^{2}r}{du^{2}}}{\left\| \frac{dr}{du}\right\|}+\frac{dr}{du}\frac{d}{du}(\ frac{1}{\left\|\frac{dr}{du}\right\|}).
Since
\frac{dt}{du}=\kappa\underline{n}
this gets me the curavture in terms of the desired bits (with n too) but I can't seem to get it to the desired result :\
Thanks for your help!