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What is the geodesic of the helicoid?

$M=\{ (x,y,z)\in \mathbb{R}^3: x\sin z - y\cos z =0\}$

Whose tangent at the point $p = (1,0,0)$ in the line $r = \{(x, y, z) \in \mathbb {R} ^ 3: x = 1, y = z\}$.

I tried with the definition, but I don't know the parameterization, and so do not know which is the tangent plane. I can't do $\alpha ''(t) \perp T_{\alpha(t)}M$.

Any help is appreciated.

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Looks pretty hard. I mean, there's an Annals paper on the subject... in which geodesics on helicoid are expressed in terms of elliptic functions. –  user53153 Dec 21 '12 at 6:00

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