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Prove: The curvature kL of the curve L at point P is related to the curvature kN 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 LN and a plane section L whose plane forms an angle θ with the plane of section LN 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 LN 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 kL of curve L is then, using Taylor’s formula, given by



where ole5.gif 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 LN has the equation z = f(x, 0) and |LN| = |fxx| . Thus


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