It’s a question of the curvature (http://planetmath.org/CurvaturePlaneCurve) of the intersection curves of the surface (1) and planes containing the -axis, which is the normal of the surface in the origin.
If the angle between the -plane and a plane containing the -axis is denoted by , when the line of intersection of the plane and the -plane is represented by the equations
then equation of the the normal section curve is
where is the abscissa and the ordinate. It follows that
thus by (2), in the origin we have
where , , the values of the derivatives ,
, in the origin.
Using those values, we obtain for the normal curvature of in the origin the value
This result gets a more illustrative form when we try to express it by using the least and the greatest value of . Instead to utilize the zeros of the derivative of the sum in (3), it’s simpler first to transfer to the double angle (http://planetmath.org/DoubleAngleIdentity),
and here to introduce an auxiliary angle () such that
This allows us to write (4) as
From this we see immediately that the curvature attains its greatest and least value when and .
Accordingly, the corresponding , 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 . One can write
So the final result, the so-called Euler’s theorem (http://planetmath.org/SecondFundamentalForm), can be expressed in the form
Here, the principal curvatures and are the greatest and the least value of the normal curvature, respectively, and is the angle between (http://planetmath.org/AngleBetweenTwoPlanes) the normal section plane corresponding and the normal section plane corresponding . As it becomes clear in the parent entry (http://planetmath.org/NormalSection), the result (6) is true not only in the origin but at any point on a surface when the given function 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).
|Date of creation||2013-03-22 17:26:27|
|Last modified on||2013-03-22 17:26:27|
|Last modified by||pahio (2872)|
|Defines||principal normal plane|