Curvature is the rate of change of direction of a curve with respect to distance along the curve. It express curve state of being curved. At any point on a curve, the curvature is the reciprocal of the radius, for other curve the curvature is the reciprocal of radius of the circle that mostly closed conforms to the curve.
Let P and Q be any neighbouring point on a curve AB such that arc AP=s and arc AQ=
so that arc PQ=
. Let the tangent to the curve at P and Q makes angle 
and 
with x-axis so that 
. Then
(1).
, measured in radius, is called the total curvature or total bending of arc PQ,
(2). The ratio
is called the average curvature of the arc PQ,
(3).
, if it exist, if it exist, is called the curvature of the curve at P and is denoted by 
.
(4). The reciprocal of curvature at any point P is called the radius of curvature and is denoted by Greek letter
The center of curvature of a curve at a point P is the point C which lies on the position direction of the normal at P and which is at a distance from it.
The circle with center C and radius CP= is called circle of curvature of the curve at P.
Any chord of the circle of curvature at P passing through P is called chord of curvature through P.
^{\frac{3}{2}}}{y_{2}}&space;\:&space;where&space;\:&space;y_{1}=\frac{\mathrm{d}&space;y}{\mathrm{d}&space;x}\:&space;\:&space;and\:&space;\:&space;y_{2}=\frac{\mathrm{d}&space;^2y}{\mathrm{d}&space;x^2} "\rho =\frac{(1+y_{1}^{2})^{\frac{3}{2}}}{y_{2}} \: where \: y_{1}=\frac{\mathrm{d} y}{\mathrm{d} x}\: \: and\: \: y_{2}=\frac{\mathrm{d} ^2y}{\mathrm{d} x^2}")
We know that the curvature of the curve at any point depends only upon its shape and so ut is independent of the coordinate system. Therefore, interchanging x and y, we get,
Definition of curvature
Let P and Q be any neighbouring point on a curve AB such that arc AP=s and arc AQ=
(1).
(2). The ratio
(3).
(4). The reciprocal of curvature at any point P is called the radius of curvature and is denoted by Greek letter
Circle, center and chord of curvature
The center of curvature of a curve at a point P is the point C which lies on the position direction of the normal at P and which is at a distance
The circle with center C and radius CP=
Any chord of the circle of curvature at P passing through P is called chord of curvature through P.
Radius of curvature
We know that the curvature of the curve at any point depends only upon its shape and so ut is independent of the coordinate system. Therefore, interchanging x and y, we get,
Center of curvature
The center of curvature for the point (x,y) is
The locus of the centre of curvature of a curve is called its evolute and the curve itself us called involute.
Example
Q1. Find the radius of curvature of the parabola 
at the point (x,y).
Sol.
The equation of the parabola is
Differentiating both side w.r.t x, we get,
Now&space;^2\right&space;]^\frac{3}{2}}{\frac{\mathrm{d}&space;^2y}{\mathrm{d}&space;x^2}} "\rho =\frac{\left [ 1+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right ) ^2\right ]^\frac{3}{2}}{\frac{\mathrm{d} ^2y}{\mathrm{d} x^2}}")
&space;^2\right&space;]^\frac{3}{2}}{\frac{-4a^2}{y^3}} "\rho =\frac{\left [ 1+\left ( \frac{4a^2}{y^2} \right ) ^2\right ]^\frac{3}{2}}{\frac{-4a^2}{y^3}}")
Sol.
The equation of the parabola is
Differentiating both side w.r.t x, we get,
Now
Q2. Prove that the curvature of a straight line is zero.
Sol.
Let the equation of the straight line be
Differentiating both side w.r.t x, we get,
= 0
Hence the result.
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