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