# Curvature of plane curves on a surface

Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?

• Does the plane contain the normal direction? – Mikhail Katz Jun 14 '16 at 17:39
• My surface is in the euclidean space. – Vincenzo Zaccaro Jun 14 '16 at 17:42
• Upper bounds in terms of what? – Igor Rivin Jun 14 '16 at 18:18
• You are more likely to have a lower bound. It seems to me that the curvature can grow a lot when the plane gets closer to the tangent. – Alex Degtyarev Jun 14 '16 at 18:22
• For Igor...in terms of the gaussian curvature and slope of the plane respect to the gauss map. – Vincenzo Zaccaro Jun 14 '16 at 22:38

Getting bounds from Gaussian curvature is hopeless due to examples like the pseudosphere, but assuming that the principal curvatures are at most $1$ one should be able to show that the curvature of the intersection should be bounded above by $\sec \alpha$ where $\alpha$ is the angle with the normal vector.