mean curvature at surface point

Let $P$ be a point on the surface$F(x,\,y,\,z)=0$  where the function $F$ is twice continuously differentiable on a neighbourhood of $P$. Then the normal curvature $\varkappa_{\theta}$ at $P$ is, by Euler’s theorem, via the principal curvatures $\varkappa_{1}$ and $\varkappa_{2}$ as

 $\displaystyle\varkappa_{\theta}=\varkappa_{1}\cos^{2}\theta+\varkappa_{2}\sin^% {2}\theta,$ (1)

where $\theta$ is the angle between (http://planetmath.org/AngleBetweenTwoPlanes) the normal section plane corresponding $\varkappa_{1}$ and the normal section plane corresponding $\varkappa_{\theta}$. When we apply (1) by taking instead $\theta$ the angle $\theta\!+\!\frac{\pi}{2}$, we may write

 $\varkappa_{\theta+\frac{\pi}{2}}=\varkappa_{1}\sin^{2}\theta+\varkappa_{2}\cos% ^{2}\theta.$

Adding this equation to (1) then yields

 $\frac{\varkappa_{\theta}+\varkappa_{\theta+\frac{\pi}{2}}}{2}=\frac{\varkappa_% {1}+\varkappa_{2}}{2}.$

The contents of this result is the

Theorem. The arithmetic mean of the curvatures (http://planetmath.org/CurvaturePlaneCurve) of two perpendicular normal sections has a value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the at the point in question.

References

• 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
 Title mean curvature at surface point Canonical name MeanCurvatureAtSurfacePoint Date of creation 2013-03-22 17:26:56 Last modified on 2013-03-22 17:26:56 Owner pahio (2872) Last modified by pahio (2872) Numerical id 8 Author pahio (2872) Entry type Theorem Classification msc 53A05 Classification msc 26B05 Classification msc 26A24 Related topic AdditionFormulasForSineAndCosine Related topic GaussianCurvature Related topic MeanCurvature Defines mean curvature