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Prove. Given a plane curve C shown in
Fig.1. Let T be the tangent to the curve at
point P, Q be a point on the curve near P
and let QM be a line perpendicular to the
tangent. Let * h* = QM and *l* = PM as
shown in the figure. Let *k* be the curvature
of the curve at point P. Then we have the
following relationship between *k*,* l* and *h*:

Proof. To prove this we choose a rectangular coordinate system in which point P lies at the origin and the x axis lies along the tangent as shown in Fig.2. We will view curve C as shown in Fig.2 as being represented by the function y = f(x). The general formula for the curvature at any selected point P on a curve y = f(x) is given by

Here in our case y' = 0 and plugging this value
into 1) gives *k* = |y''| . Let us now expand the
function y = f(x) about the point x = 0 using
Taylor’s Formula. We get

y = ½ y''x^{2} + ε

where ε is an infinitesimal of higher order that can be neglected. Thus

and since |y| = *h* and x^{2} = *l*^{2} we have

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