PlanetMath: mean curvature at surface point

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mean curvature at surface point

(Theorem)

Let be a point on the surface $F(x,\,y,\,z) = 0$ where the function is twice continuously differentiable on a neighbourhood of . Then the normal curvature $\varkappa_\theta$ at is, by Euler's theorem, expressible 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 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

$\displaystyle \varkappa_{\theta+\frac{\pi}{2}} = \varkappa_1\sin^2\theta+\varkappa_2\cos^2\theta.$

Adding this equation to (1) then yields

$\displaystyle \frac{\varkappa_\theta+\varkappa_{\theta+\frac{\pi}{2}}}{2} = \frac{\varkappa_1+\varkappa_2}{2}.$

The contents of this result is the

Theorem. The arithmetic mean of the curvatures of two perpendicular normal sections has a constant value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the mean curvature at the point in question.

Bibliography

1: ERNST LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).

"mean curvature at surface point" is owned by pahio.

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Also defines:

mean curvature

Keywords:

normal section, curvature

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Attachments:: minimal surface (Definition) by pahio

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Cross-references: perpendicular, arithmetic mean, equation, angle, plane, normal section, principal curvatures, Euler's theorem, normal curvature, neighbourhood, continuously differentiable, function, surface, point
There are 4 references to this entry.

This is version 5 of mean curvature at surface point, born on 2007-08-03, modified 2007-08-05.
Object id is 9830, canonical name is MeanCurvatureAtSurfacePoint.
Accessed 788 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

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