normal section[′nȯr·məl ′sek·shən]
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 k1 and k2 (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 kn of the normal section in a direction forming an angle ϕ with the first principal direction is related to k1 and k2 by the expression
kn = k1 cos2 ϕ + k2 sin2 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 kn is the normal curvature of the surface in the direction of line a.