mean curvature at surface point

Let P be a point on the surfaceF(x,y,z)=0  where the function F is twice continuously differentiable on a neighbourhood of P. Then the normal curvatureMathworldPlanetmathPlanetmathPlanetmath ϰθ at P is, by Euler’s theorem, via the principal curvatures ϰ1 and ϰ2 as

ϰθ=ϰ1cos2θ+ϰ2sin2θ, (1)

where θ is the angle between ( the normal sectionPlanetmathPlanetmath plane corresponding ϰ1 and the normal section plane corresponding ϰθ. When we apply (1) by taking instead θ the angle θ+π2, we may write


Adding this equation to (1) then yields


The contents of this result is the

Theorem. The arithmetic meanMathworldPlanetmath of the curvaturesMathworldPlanetmathPlanetmath ( of two perpendicularPlanetmathPlanetmath normal sections has a value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the mean curvatureMathworldPlanetmathPlanetmathPlanetmath at the point in question.


  • 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