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

${\varkappa }_{\theta }={\varkappa }_1{\cos }^2\theta +{\varkappa }_2{\sin }^2\theta ,$ (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 mean curvature at the point in question.

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