Let $\gamma$ be a unit-speed curve on the helicoid $$\sigma (u,v)=(u\cos v, u\sin v, v)$$ I have shown that $$\dot u^2+(1+u^2)\dot v^2=1$$ and that if $\gamma$ is a geodesic on $\sigma$ then $$\dot v=\frac{a}{1+u^2}$$ where $a$ is a constant.
Could you give me a hint how we could find the geodesics corresponding to $a = 0$ and $a = 1$ ?