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It's given in the question that $\kappa=\tau=1/s$.

By Serret-Frenet Formulae I know that

$\vec{r\prime}=\frac{1}{s}\hat{n}$ and $\vec{r\prime\prime}=\frac{-1}{s^2}\hat{n}+\frac{1}{s}(\frac{1}{s}\hat{b}-\frac{1}{s}\hat{t})=\frac{-1}{s^2}\hat{n}+\frac{1}{s^2}\hat{b}-\frac{1}{s^2}\hat{t}$

$\hat{t},\hat{n},\hat{b}$ are functions of s and their components are unknown. How can I go further to solve the question?

Asit Srivastava
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1 Answers1

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It'll be a conical helix, namely

\begin{align*} \mathbf{r}(s) &= \begin{pmatrix} \dfrac{s}{\sqrt{6}} \, \cos (\sqrt{2}\, \ln s) \\[2pt] \dfrac{s}{\sqrt{6}} \, \sin (\sqrt{2}\, \ln s) \\[2pt] \dfrac{s}{\sqrt{2}} \end{pmatrix} \\[5pt] \kappa &= \frac{1}{s} \\[5pt] \tau &= \frac{1}{s} \end{align*}

For more general case, see another answer here and also the ODE for space curve here.

Ng Chung Tak
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