# Differentiate curvature, torsion

I want to construct a frenet curve $Y: \Bbb R \to \Bbb R^3$ with constant curvature $K$ and torsion $T$. I figured I'd start with calculating the derivatives of $K$ and $T$ but I don't know how to differentiate those expressions. They are specifically,

$$K = \|(Y' \times Y'')\|/\|Y'\|^3,$$

$$T = \det[Y', Y'', Y''']/\|Y' \times Y''\|^2.$$

Thanks for your time!

• You mean you want to differentiate the two equations, equating them to zero and solving/integrating them together? – Narasimham Sep 12 '17 at 15:33
• Yes, somehow. I don't know how it would work out though, but that was my initial stance. A commenter suggested using a theorem I don't know of, but I googled and a "Picard-Lindelöf" theorem showed up. In either case, I still want to learn how to differentiate such expressions.. – iaenstrom Sep 12 '17 at 16:38

## 1 Answer

You might want to consider exploiting the existence and uniqueness theorem for such curves, given $K$ and $T$. By the theorem, it suffices to exhibit a curve with the desired properties, and every other curve will have to be congruent to the one exhibited. You could start with the helix $\alpha(t) = (a \cos \omega t, a \sin \omega t, bt)$.

The Frenet-Serret equations and the fundamental theorem of the theory of curves can be found in most introductory textbooks in classical differential geometry. For example you can consult the book by do Carmo "differential geometry of curves and surfaces" where the proof is given on pages 309-311.

• Thanks for your answer! Are you referring to the Picard Lindelöf theorem? If not, which one are you referring to? I understand your stance and solution conceptually but is it necessary to apply the theorem your referring to or does it suffice just to refer to it? Thanks again! – iaenstrom Sep 12 '17 at 16:40
• As far as the course policy is concerned you should consult with the lecturer. – Mikhail Katz Sep 13 '17 at 6:57