# normal section

Let $P$ be a point of a surface

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

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 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

 $\displaystyle z=z(x,\,y),$ (2)

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

 $z(0,\,0)=z^{\prime}_{x}(0,\,0)=z^{\prime}_{y}(0,\,0)=0.$

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 plane of the surface (1) at $P$. The equation (1) defines the function of (2).

Title normal section NormalSection 2013-03-22 17:26:25 2013-03-22 17:26:25 pahio (2872) pahio (2872) 10 pahio (2872) Definition msc 53A05 msc 26A24 msc 26B05 SecondFundamentalForm DihedralAngle normal curvature