# Finding the angle between 2 curves on a surface.

I have the following question:

Let $$M\subset R^3$$ be a regular surface and $$A\in M$$ a point where the gauss curvature $$K$$ and the mean curvature $$H$$ satisfy $$K\neq 0, H=0$$. Also, let $$\alpha, \beta:(-1,1)\to M$$ be regular curves such that $$\alpha(0)=\beta(0)=A$$ and the angle between their tangents at this point is not $$0$$. Also, $$\alpha''(0),\beta''(0)\in T_AM$$. Find the angle between the curves in $$A$$.

I can conclude from the condition on the curvatures that $$K<0$$. I am also familiar with the first and second fundamental forms, sectional curvature and curvature operator and "theorema egregium", Though I can't seem to find a direction to a solution.

HINT: The conditions you were given on the second derivatives tell you what the normal curvatures of the curves are at $$A$$. There is a famous formula for the normal curvature in the direction $$\theta$$ relative to the principal directions; use it! (This question has appeared numerous times on this site, once you realize there's a word for these curves ...)