# Normal Section

## normal section

[′nȯr·məl ′sek·shən]## Normal Section

A normal section of a surface *S* at a given point *M* on the surface is the curve of intersection of *S* with a plane drawn through the normal at the point *M*. Normal sections are used to study the curvature of *S* in different (tangential) directions at *M*. Among these there are two (mutually perpendicular) directions, called principal, for which the normal curvature, that is, the curvature of the corresponding normal section, attains its maximum and minimum values *k*_{1} and *k*_{2} (the principal curvatures at the given point). The curvature of a normal section is assigned a plus (minus) sign if the direction of concavity of the section coincides with (is opposite to) the positive direction of the normal to the surface. The normal curvature of a surface in a given direction can be expressed very simply in terms of the principal curvatures; specifically, the curvature *k _{n}* of the normal section in a direction forming an angle ϕ with the first principal direction is related to

*k*

_{1}and

*k*

_{2}by the expression

*k _{n}* =

*k*

_{1}cos

^{2}ϕ +

*k*

_{2}sin

^{2}ϕ

which is Euler’s theorem on normal curvature. The curvatures of oblique sections of a surface are also studied by using the curvatures of normal sections. Specifically, the curvature *k* of an oblique section of a plane a that passes through a given tangent line *a* is expressed by Meusnier’s formula

where ϕ is the angle between the plane *α* and the normal to the surface and *k _{n}* is the normal curvature of the surface in the direction of line

*a*.