Parametrising a curve using curvature and torsion functions

Question

I am trying to get a parametrization of the curve whose curvature and torsion functions are given as

$$\kappa(s)= \dfrac{1}{1+s^2} ,\;\; \tau(s) = \dfrac{s}{1+s^2}$$

I know that in general it is not possible to get parametrizations from the curvature and torsion functions, but I was hoping this one would fit the bill.

The curvature function is that of a catenary curve, so I thought a parametrization would be along the lines of

$$ x(s) = \int \cos(\arctan(s))\mathsf{ds}, \;\; y(s) =\int \sin(\arctan(s))\mathsf{ds}, \;\; z(s) = ?$$ Am I on the right track??

Further the curvature and torsion functions indicate that this a geodesic on a cone. But I am not able to push on from here.

score 1 · Answer 1 · answered Feb 11 '16 at 11:09

1

Cone geodesic should be

$$ \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \sqrt{a^{2}+s^{2}} \left( \begin{array}{c} \sin \beta \cos \left( \csc \beta \tan^{-1} \frac{s}{a} \right) \\ \sin \beta \sin \left( \csc \beta \tan^{-1} \frac{s}{a} \right) \\ \cos \beta \end{array} \right)$$

but with \begin{align*} \kappa &= \frac{a^{2}\cot \beta}{(a^{2}+s^{2})^{\frac{3}{2}}} \\ \tau &= \frac{as\cot \beta}{(a^{2}+s^{2})^{\frac{3}{2}}} \end{align*}

instead.

answered Feb 11 '16 at 11:09

Ng Chung Tak

18,990

Well, I thought any geodesic on a cone should satisfy the relation that $\dfrac{d}{ds}(\dfrac{\tau}{\kappa})$ is a constant. Am I right? – Vishesh Feb 11 '16 at 12:10
Your curve also does that. But I wished to know if the curve i mentioned is also a geodesic on a cone possibly. – Vishesh Feb 11 '16 at 12:15
Unlikely. The intrinsic equations seem to be invariant under the scaling of $a$ for fixed $\beta$. May your case is for other kind of surface. – Ng Chung Tak Feb 11 '16 at 14:00
Ok and what about the parametrisation of the curve?Can that be found? – Vishesh Feb 11 '16 at 14:25
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Maybe easier than the following : http://math.stackexchange.com/questions/1603719/what-is-the-space-curve-with-curvature-and-torsion-obeying – Narasimham Feb 12 '16 at 18:23
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This may help http://www.emis.de/journals/BMAA/repository/docs/BMAA3-2-19.pdf – Ng Chung Tak Feb 14 '16 at 06:41
Thanks to both you guys for those links – Vishesh Feb 18 '16 at 07:48

Parametrising a curve using curvature and torsion functions

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