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I am trying to find a proof for the 9th question of section 2.4, from the book Elementary Differential Geometry by Barrett O'Neill. I want to show that a curve $\alpha$ with curvature $\kappa$ and torsion $\tau$ both constant is a circular helix.

My thoughts: I know $\alpha$ must be a cylindrical helix since $\tau/\kappa$ is a constant. Remains to show that it is circular. I'm not sure how to do this. I found this question here : Is the helix the unique path with constant curvature and constant torsion? where the answer gives a hint to prove N′′(s)=−(κ2+τ2)N(s). I can do this, but where does it head me toward? I cannot see where this will lead me.

Random thought: It occurs to me that if I can project the curve $\alpha$ on the xz plane, it must be a circle. But I don't know how I would go about showing this.

Any help is appreciated. Thank you for your time.

1 Answers1

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Of course the uniqueness theorem (Fundamental Theorem of Curves) will do it, since you know (or can easily show) that circular helices have constant curvature and torsion.

The intent of my hint was to then deduce that $\vec N(s)=\cos(ks)\vec c_1+\sin(ks)\vec c_2$ with $k^2=\kappa^2+\tau^2$ and $\vec c_1,\vec c_2$ fixed vectors. Show then that $\vec c_1$ and $\vec c_2$ must be orthogonal and show that you get circular motion in the plane they span. Can you finish from there?

Ted Shifrin
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  • I will try and come back to comment. Thanks a lot for the answer. –  Sep 27 '17 at 05:54
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    Sure thing. If you get very stuck and/or have further questions, let me know. When you figure it out, accept the answer so that the question doesn't stay on the unanswered list. – Ted Shifrin Sep 27 '17 at 06:01
  • How do I prove $\vec c_1$ and $\vec c_2$ are orthogonal? I applied the Frenet formula $\vec T=\kappa \vec N$. I substitue the value of $\vec N(s)=\cos(ks)\vec c_1+\sin(cs)\vec c_2$ with $k^2=\kappa^2+\tau^2$. Then integrate for $\vec T$. then i took dot product between $\vec{N}$ and $\vec{T}$. I am not getting zero. I am getting complicated form. – Unknown x Aug 02 '23 at 13:22
  • Could you give hint? – Unknown x Aug 02 '23 at 13:23
  • @Unknownx Consider $\vec N(0)$ and $\vec N’(0)$. Apparently, I made a typo in this answer years ago. I just edited. – Ted Shifrin Aug 02 '23 at 15:10
  • I got $\vec{N}(0)=\vec{c}_1$ and $\vec{N}'(0)=-\vec{c}_2=-\kappa(0)\vec{T}(0)+\tau(0)\vec{B}(0)$. Which results $ \vec{c}_1\cdot \vec{c}_2=0$ – Unknown x Aug 02 '23 at 18:21
  • There is a small error in your derivative, but the conclusion is correct. Also, remember that $\vec f \cdot \vec f’=0$ whenever $\vec f$ has constant length. @Unknownx – Ted Shifrin Aug 02 '23 at 18:25
  • Yes. Thank you for pointing out my mistake. I got $\vec{c}_1$ and $\vec{c}_2$ are orthogonal. How does it create a circular motion? – Unknown x Aug 02 '23 at 18:38
  • Using $T'(s)=\kappa N(s)$. I integrate to obtain $T(s)$. Then I took dot product between $N(s)$ and $T(s)$. I am not getting zero. I am getting $T(s)\cdot N(s)=(\kappa/k)\sin(ks)\cos(ks) (||c_1||^2-||c_2||^2)$ – Unknown x Aug 03 '23 at 07:58
  • I know $||c_1||^2=1$ – Unknown x Aug 03 '23 at 08:02
  • @Unknownx So what is $\alpha(s)$? – Ted Shifrin Aug 03 '23 at 18:43
  • Integration of $T(s)$. Right? – Unknown x Aug 04 '23 at 14:45