0

I have a basic question regarding the definition of a curvature.

Most of my searches revealed the following:

κ = -T⋅$dN/ds$,

where T is the tangent vector, N is the normal vector, s is the arc length, k is the curvature.

I use a CFD code where the normal vector is computed and for the curvature the following equation is used:

$k = -divergence.N$, This equation also makes it way into Wiki Wikipedia curvature calculation unfortunately without proof.

Can anyone help me understand how the 1st and 2nd equations are related.

Saideep
  • 11

1 Answers1

1

You seem to be totally mixing things.

On the one hand, $\kappa = - \textrm T \cdot \dfrac {\Bbb d \textrm N} {\Bbb d s}$ is obtained from the second Frenet formula by taking the scalar product with $\textrm T$. It is about the curvature of a curve.

On the other hand, $H = -\frac 1 2 \text{div } N$ is just another way of computing the mean curvature of a hypersurface.

There is no connection between these two types of curvatures, let alone the fact that there exist other types of curvature as well.

Alex M.
  • 35,207
  • Ah, thanks for pointing out. Did not realize that. I am looking at fluid interface curvature. Any idea about a proof related to k = -divergence(N)? – Saideep Sep 03 '16 at 11:46
  • @Well, in order to answer that, you should first specify how you define the mean curvature. If you are only interested in orientable submanifolds of codimension $1$, then you could simply take that formula as a definition. – Alex M. Sep 03 '16 at 11:54
  • Thanks you directed me. I found the following paper that proves what I was asking for. It is related to Youngs-Laplace equations.http://willson.cm.utexas.edu/Research/Sub_Files/Surface_Phenomena/Spring%202000/surface_normal_proof.pdf – Saideep Sep 03 '16 at 13:27