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# What is curvature and radius of curvature?

## Overview

The degree of curvedness is used to determine the shape of a plane curve. Curvature is nothing but the measure of rate of change in of curvedness.

Here note that there is no bending in a straight line while in a circle there is constant bending.

Angle of contingence of the arc AB of a curve c is the angle between the tangents at A and B to the curve c

Given two arcs of the same length, the arc with greater angle of contingence is said to be more curved.

## How do we find it?

Let the equation of the curve is given in Cartesian form y = f(x), then

Formula for polar form-

Radius of curvature is the reciprocal of curvature at any point. It is denoted by (rho), then

Or

Parametric form:

Let the co-ordinates be defined in the form of functions depend on one independent variable t.

Suppose

, Then we will calculate the values in term of variables depend on t.

By chain rule we have

And

By

Newton’s formula

a.      If x-axis is tangent  to a curve  at the origin, then

b.     If y-axis is tangent  to a curve  at the origin, then

c.      When  the tangent is  at the origin and neither  on x-axis nor on  y-axis then

Where .

Where

Centre of curvature-

Centre of curvature at any point P (x, y) on the curve y = f(x) is given by-

Circle of curvature at P-

Example: Find the radius of curvature at (0, c) of the catenary y = c cosh(x/c)?

Solution:

Given catenary

Differentiate (i) with respect to x.

Also consider

Again differentiating with respect to x , we get

We know that

Substituting values from (i),(ii) and (iii) we get

Hence

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