# Frenet frame

1. Nov 7, 2012

### BitterX

1. The problem statement, all variables and given/known data

calculate the frenet frame for the vector:
$\vec{r}(t)=(2t cost,2tsint,5t)$

calculate the acceleration in frenet frame.

2. Relevant equations
$\hat{T}=\frac{dr}{ds}=\frac{\dot{r}}{|\dot{r}|}$
$\hat{N}=\frac{\frac{dT}{ds}}{|\frac{dT}{ds}|}$
$\hat{B}=\hat{T}\times \hat{N}$

3. The attempt at a solution

I'm not too sure how to get to $\vec{r}(s)$
what I tried is : $s=2t \ \ \Rightarrow \vec{r}(s)= s(\cos \frac{s}{2},\sin \frac{s}{2},2.5)$
$T=\frac{dr}{ds}=s(-\sin \frac{s}{2},\cos \frac{s}{2},0)+(\cos \frac{s}{2},\sin \frac{s}{2},2.5)$

which is NOT a unit vector.
if I try use t as the variable it's just becomes a gigantic answer.

2. Nov 7, 2012

### micromass

Staff Emeritus
There are two ways to solve this problem:

• The first way is to put r into arclength parametrization. You should have seen several formulas in your course that show how to make r into a curve with unit speed. These formulas use integrals, so are not always handy or solvable, however.
• The second way is to use formulas for the Frenet frame which do not require the curve to have unit speed. These formulas are a bit more complicated than the usual formulas.

So, look in your course for these things and see if you can do something with it.

3. Nov 8, 2012

### BitterX

Thanks.
I know the second way,and I did it like that.
About the first way, can you recommend on a book that show how to do it?
in our course it was only when z=0...