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Prove: The curvature k_{L} of the
curve L at point P is related to
the curvature k_{N} of the normal
section of the same direction (i.e.
plane Q and the plane of the
normal section intersect the
tangent plane in the same straight
line) at P by the formula

Proof. Let us consider, at point P, a normal section L_{N} and a plane section L whose plane forms
an angle θ with the plane of section L_{N} as shown in Fig. 1. The x and y axes lie in the tangent
plane and we take the x axis to be tangent to the curves L_{N} and L at the origin. Let X be a point
on curve L as shown in the figure. Its coordinates are (x, y, f(x, y)). The perpendicular distance
from X to the x axis is h. It can be seen that h is a function of x and y according to

The curvature k_{L} of curve L is then, using Taylor’s formula, given by

where Since the x axis is tangent to the curve L,

Thus, on taking the limit in 5), we get

Now for the chosen coordinate system the curve L_{N} has the equation z = f(x, 0) and |L_{N}| = |f_{xx}| .
Thus

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