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).