Goseeko blog

What is curvature and radius of curvature?

by Team Goseeko


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

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


Parametric form:

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


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

By chain rule we have



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 .


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)?


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


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