mean curvature at surface point
Let be a point on the surface where the function is twice continuously differentiable on a neighbourhood of . Then the normal curvature at is, by Euler’s theorem, via the principal curvatures and as
(1) |
where is the angle between (http://planetmath.org/AngleBetweenTwoPlanes) the normal section plane corresponding and the normal section plane corresponding . When we apply (1) by taking instead the angle , we may write
Adding this equation to (1) then yields
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.
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 |