PlanetMath: normal section

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normal section

(Definition)

Normal sections

Let be a point of a surface

$\displaystyle F(x,\,y,\,z) = 0,$

(1)

where

has the continuous first and second order partial derivatives in a neighbourhood of

. If one intersects the surface with a plane containing the surface normal at

, the intersection curve is called a normal section.

Normal curvatures

When the direction of the intersecting plane is varied, one gets different normal sections, and their curvatures at , the so-called normal curvatures, vary having a minimum value $\varkappa_1$ and a maximum value $\varkappa_2$ . The arithmetic mean of $\varkappa_1$ and $\varkappa_2$ is called the mean curvature of the surface at .

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

$\displaystyle z = z(x,\,y),$

(2)

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

$\displaystyle z(0,\,0) = z'_x(0,\,0) = z'_y(0,\,0) = 0.$

Indeed, one can take a new rectangular coordinate system with

the new origin and the normal at

the new

-axis; then the new

-plane coincides with the tangent plane of the surface (1) at

. The equation (1) defines the function of (2).

"normal section" is owned by pahio.

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See Also: second fundamental form

Also defines:

normal curvature

This object's parent.

Attachments:: normal curvatures (Topic) by pahio

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Cross-references: equation, tangent plane, normal, rectangular coordinate, origin, without loss of generality, function, mean curvature, arithmetic mean, normal curvatures, curve, surface normal, plane, intersects, neighbourhood, partial derivatives, continuous, surface, point
There are 4 references to this entry.

This is version 7 of normal section, born on 2007-07-30, modified 2007-07-30.
Object id is 9820, canonical name is NormalSection.
Accessed 760 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)

Pending Errata and Addenda

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