Normal curvature

Given a regular surface and a curve within that surface, the normal curvature at a point is the amount of the curve's curvature in the direction of the surface normal. The curve on the surface passes through a point $ p$ , with tangent $ v$ , curvature $ k$ and normal $ n$ . Given the surface normal $ N$ , the normal curvature $ k_n$ is the length of the projection of $ kn$ onto $ N$ , namely $ \ensuremath{\langle kn, N \rangle}$ .

If we cut a plane containing both $ N$ and $ v$ through the surface, we generate a second curve on the surface that agrees with the first curve at $ p$ . The normal curvature is the same as the curvature of this second curve at $ p$ . In addition, the Meusnier theorem states that the curves generated by cutting any plane that contains both $ p$ and $ v$ all have the same normal curvature.


Copyright © 2005 Adrian Secord. All rights reserved.