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.
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.