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 , with tangent , curvature and normal . Given the surface normal , the normal curvature is the length of the projection of onto , namely .
If we cut a plane containing both and through the surface, we generate a second curve on the surface that agrees with the first curve at . The normal curvature is the same as the curvature of this second curve at . In addition, the Meusnier theorem states that the curves generated by cutting any plane that contains both and all have the same normal curvature.
Copyright © 2005 Adrian Secord. All rights reserved.