normal section

Normal sections

Let $P$ be a point of a surface

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

where $F$ has the continuous first and partial derivatives 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.

Normal curvatures

When the direction of the intersecting plane is varied, one gets different normal sections, and their curvatures (http://planetmath.org/CurvaturePlaneCurve) at $P$, 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 $P$.

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

$$z(0,\mathrm{\hspace{0.17em}0})=z_x^{\prime }(0,\mathrm{\hspace{0.17em}0})=z_y^{\prime }(0,\mathrm{\hspace{0.17em}0})=0.$$

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 plane 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