Curvature of a space curve

1. Jul 15, 2009

kidsmoker

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

$$\kappa=\frac{\left\|\underline{\dot{r}}\times\underline{\ddot{r}}\right\|}{\left\|\underline{\dot{r}}\right\|^{3}}$$ .

3. The attempt at a solution

We have

$$t(u)=\frac{\frac{dr}{du}}{\left\|\frac{dr}{du}\right\|}$$

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!

2. Jul 15, 2009

n!kofeyn

Last edited by a moderator: Apr 24, 2017
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