Definition of curvature

Circle, center and chord of curvature

Radius of curvature

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 , measured in radius, is called theorof arc PQ,The ratio is called theof the arc PQ, , if it exist, if it exist, is called theof the curve at P and is denoted by (4). The reciprocal of curvature at any point P is called theand 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 calledof the curve at P.Any chord of the circle of curvature at P passing through P is calledthrough P.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,