normal section

Let P be a point of a surface

F(x,y,z)=0, (1)

where F has the continuousMathworldPlanetmath first and partial derivativesMathworldPlanetmath in a neighbourhood of P.  If one intersects the surface with a plane containing the surface normal at P, the intersection curve is called a normal section.

When the direction of the intersecting plane is varied, one gets different normal sections, and their curvaturesMathworldPlanetmathPlanetmath ( at P, the so-called normal curvatures, vary having a minimum value ϰ1 and a maximum value ϰ2.  The arithmetic meanMathworldPlanetmath of ϰ1 and ϰ2 is called the mean curvatureMathworldPlanetmathPlanetmathPlanetmath of the surface at P.

By the suppositions on the function F, examining the normal curvatures can without loss of generality be to the following:  Examine the curvature of the normal sections through the origin, the surface given in the form

z=z(x,y), (2)

where  z(x,y)  has the continuous first and partial derivatives in a neighbourhood of the origin and

z(0, 0)=zx(0, 0)=zy(0, 0)=0.

Indeed, one can take a new rectangular coordinate system with P the new origin and the normal at P the new z-axis; then the new xy-plane coincides with the tangent planeMathworldPlanetmath of the surface (1) at P. The equation (1) defines the function of (2).

Title normal section
Canonical name NormalSection
Date of creation 2013-03-22 17:26:25
Last modified on 2013-03-22 17:26:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 53A05
Classification msc 26A24
Classification msc 26B05
Related topic SecondFundamentalForm
Related topic DihedralAngle
Defines normal curvature