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Homenormal section

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# normal section

Normal sections

Let $P$ be a point of a surface

$\displaystyle F(x,\,y,\,z)=0,$ | (1) |

where $F$ has the continuous first and second order 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 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 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

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

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).

## Mathematics Subject Classification

53A05*no label found*26A24

*no label found*26B05

*no label found*

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