Differential geometry Frenet

1. Jul 4, 2014

ParisSpart

even γ: I-> R ^ 2 curve parameterized as to arc length (single speed) with curvature k (s)> 0 and torsion τ(s)> 0. I want to write the γ(s) as a combination of n(s), t(s), b(s). these are the types of Frenet.

the only thing i know is that the types of Frenet are t(s)=γ'(s) , b(s)=t(s)Xn(s) and n(s)=t'(s)/k(s)
but how i will write γ(s) with this types? is there any way or type to do this?

2. Jul 5, 2014

Maybe it could help if you told us what you are trying to do. Are you trying to solve a specific problem? Also, did you intend to write $\gamma : I \rightarrow \mathbb{R}^2$, as in a planar curve?

3. Jul 5, 2014

ParisSpart

my problem is:even γ: I-> R ^3 curve parameterized as to arc length (single speed) with curvature k (s)> 0 and torsion τ(s)>0. we assume that the γ is at the surface sphere with center the origin. Show that for any s we have:
γ(s)=-(1/k(s))*n(s) + (k'(s)/(k^2(s)*τ(s)))*b(s)

and he gives us a hint: write the γ(s) as a combination of n(s), t(s), b(s). these are the types of Frenet
and γ:I->R^3

4. Jul 5, 2014

ParisSpart

i think that if i find a way to combine γ with the types of frenet and derivative them i will find the problem

5. Jul 5, 2014

You are meant to write $\gamma(s) = a_1(s)t(s) + a_2(s)n(s) + a_3(s)b(s)$, for unknown scalar functions $a_i(s)$. You can immediately conclude that one of the $a_i(s)$ is zero identically. Which one and why? The other two can be determined by differentiating the remaining expression.

6. Jul 5, 2014

ParisSpart

a1(s) will be zero? and after that with differentiation i will conclude to the the expression i wrote?

7. Jul 5, 2014

Don't ask. Try it! And if you run into more trouble, come back here, show us what you have done and we will give you more help.

8. Jul 5, 2014

ParisSpart

i tried to the derivative and i found this :
γ'(s)=a2(s)n'(s)+a2'(s)n(s)+a3'(s)b(s)+a3(s)b'(s) , we know that γ'(s)=0 beacuse we are on the surface of sphere , after that i tried to replace n'(s)=-k(s)t(s)+τ(s)b(s) and the other types?

9. Jul 5, 2014

micromass

Staff Emeritus
Why will $a_1(s)$ be zero?

Why would $\gamma^\prime(s) = 0$? I don't see what the surface of the sphere has to do with this.

Do you know the Frenet formulas?

10. Jul 5, 2014

ParisSpart

because γ(s)*γ(s)=c where c is a constant and then 2γ(s)γ'(s)=0

11. Jul 5, 2014

micromass

Staff Emeritus
OK, how does that answer my two questions?

12. Jul 5, 2014

$\gamma'(s) = 0$ identically would mean that there is no motion and the curve consists of just one point. Clearly this is not true in our case, since we are allowed to move around as long as we stay on the surface of the sphere.

13. Jul 5, 2014

ParisSpart

i think that a1(s)=0 because i supposed that for γ'(s)=0 we know that t(s)=γ'(s) and then t(s)=0 and a1(s)=0

14. Jul 5, 2014

ParisSpart

but if we dont have γ'(s)=0 if we differentiate the γ(s)=a1(s)t(s)+a2(s)n(s)+a3(s)b(s) with what γ'(s) will be equal?

15. Jul 5, 2014

That is not correct and it is not the reason $a_1(s)$ is zero. You wrote above that $\gamma(s) \cdot \gamma'(s) = 0$, which is correct. Do you see how you could use this to conclude that $a_1(s)$ must be zero?

16. Jul 5, 2014

ParisSpart

no... can you tell me why? i focused on this that i said prior

17. Jul 5, 2014

So $\gamma(s) = a_1(s)t(s) + a_2(s)n(s) + a_3(s)b(s)$ and $\gamma'(s) \cdot \gamma(s) = 0$, where $\gamma'(s) = t(s)$ by definition.

By the way, I hope you understand that $\gamma(s)$ and $\gamma'(s)$ are vector-valued: $\gamma(s) = (\gamma_1(s), \gamma_2(s), \gamma_3(s)), \, \gamma'(s) = (\gamma'_1(s), \gamma'_2(s), \gamma'_3(s))$ and that by $\gamma(s) \cdot \gamma'(s)$ I mean the scalar product of these vectors.

18. Jul 6, 2014

ParisSpart

if i differentiate i find this γ'(s)=a2(s)n'(s)+a2'(s)n(s)+a3'(s)b(s)+a3(s)b'(s) beacuse γ'(s)=t(s) i will replace this in this equation? and after that i am trying to replace the followings:
b(s)=t(s)xn(s) , n(s)=t'(s)/k(s), t'(s)=k(s)n(s), n'(s)=-k(s)t(s)+τ(s)b(s), b'(s)=-τ(s)n(s) ?

19. Jul 6, 2014

Use the last two of these and then scalar multiply the expression $\gamma'(s) = ...$ with some suitable vector.