normal curvatures

Let us determine the normal curvaturesMathworldPlanetmathPlanetmathPlanetmath ( ϰ of the surface

z=z(x,y) (1)

in the origin, when (1) has the continuousMathworldPlanetmath 1st and 2nd order partial derivativesMathworldPlanetmath in a neighbourhood of  (0, 0)  and satisfies

z(0, 0)=zx(0, 0)=zy(0, 0)=0. (2)

It’s a question of the curvatureMathworldPlanetmathPlanetmath ( of the intersection curves of the surface (1) and planes containing the z-axis, which is the normal of the surface in the origin.

If the angle between the zx-plane and a plane τ containing the z-axis is denoted by φ, when the line of intersection of the plane τ and the xy-plane is represented by the equations

x=ϱcosφ,y=ϱsinφ  (-<ϱ<),

then equation of the the normal sectionMathworldPlanetmath curve Cφ is


where ϱ is the abscissa and z the ordinate.  It follows that


thus by (2), in the origin we have

dzdϱ= 0,d2zdϱ2=acos2φ+2bsinφcosφ+csin2φ,

where a, b, c the values of the derivatives 2zx2, 2zxy, 2zy2 in the origin.

Using those values, we obtain for the normal curvature of Cφ in the origin the value

ϰ(φ)=[d2zdϱ2(1+(dzdϱ)2)3/2]ϱ= 0=acos2φ+2bsinφcosφ+csin2φ. (3)

This result gets a more illustrative form when we try to express it by using the least and the greatest value of ϰ(φ).  Instead to utilize the zeros of the derivative of the sum in (3), it’s simpler first to transfer to the double angle (,

ϰ(φ)=a+c2+a-c2cos2φ+bsin2φ, (4)

and here to introduce an auxiliary angle α  (0α<π) such that


This allows us to write (4) as

ϰ(φ)=a+c2+kcos2(φ-α). (5)

From this we see immediately that the curvature attains its greatest and least value a+c2±k when  φ=α  and  φ=α+π2.

Accordingly, the corresponding τ, the principal normal planes, are perpendicular to each other; their normal section curves on the surface (1) in the origin are briefly called the principal sections.

The expression (5) of the normal curvature may still be edited.  Let us take a new parameter angle  φ-α:=θ.  One can write


So the final result, the so-called Euler’s theorem (, can be expressed in the form

ϰθ=ϰ1cos2θ+ϰ2sin2θ. (6)

Here, the principal curvatures ϰ1 and ϰ2 are the greatest and the least value of the normal curvature, respectively, and θ is the angle between ( the normal section plane corresponding ϰ1 and the normal section plane corresponding ϰθ. As it becomes clear in the parent entry (, the result (6) is true not only in the origin but at any point on a surface when the given functionMathworldPlanetmath has the continuous 1st and 2nd derivatives in some neighbourhood of the point.


  • 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset II.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
Title normal curvatures
Canonical name NormalCurvatures
Date of creation 2013-03-22 17:26:27
Last modified on 2013-03-22 17:26:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Topic
Classification msc 53A05
Classification msc 26B05
Classification msc 26A24
Related topic SecondFundamentalForm
Related topic MeusniersTheorem
Related topic ErnstLindelof
Defines principal normal plane
Defines principal section
Defines principal curvature
Defines Euler’s theorem