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.

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