Normal curvature of helicoid

Question

Question

Let $S$ be a helicoid $\sigma(u, v) = (u \cos v, u \sin v, v)$ Find the normal curvature in the direction of a general unit tangent vector at the origin $p = (0, 0, 0)$

What I've done

What's next?

It does not sound like a final answer to me - what else can I do?

For instance, should I substitute

$\dot{v} = \pm \sqrt{1 - \dot{u}^2}$ and write the answer as $k_N = \pm 2 \dot{u}\sqrt{1 - \dot{u}^2}$

Answer 1

You're right. The general unit tangent vector is given by specifying $\dot u$ and $\dot v$ as you said. The helicoid has the property that the $u$- and $v$-curves are both asymptotic curves (this is obvious for the lines, less obvious for the helices), and so $L=N=0$ actually holds at every point of the helicoid.