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ole.gif

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

                                                

             ole1.gif




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

 

             ole2.gif  


The curvature kL of curve L is then, using Taylor’s formula, given by


             ole3.gif


ole4.gif



where ole5.gif Since the x axis is tangent to the curve L,


             ole6.gif


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


             ole7.gif


Now for the chosen coordinate system the curve LN has the equation z = f(x, 0) and |LN| = |fxx| . Thus

             ole8.gif




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