Curvature:

Curvature defined as bending. You may consider that the turn on a 25 paise coin shows a more rapid change of way per unit of arc length than that on a one rupee coin. Mathematically saying, we consider that the curvature at a point on a 25 paise coin is bigger than that at the corresponding coordinate on a rupee coin. Now we provide the proper definitions of the curvature of a curve at a point.

Curvature of a curve at a point

Definition: Suppose A be a fixed point on a curve. Let Q and P be two neighboring points on the curve such that

AP = s and AQ = s + Δs so that PQ = Δs.

 Let the tangents at Q and P creates angles ψ and ψ + Δ ψ respectively with the positive direction of the x-axis. Then Δ ψ is the angle through which the tangent revolves from P and Q. The ratio  is known as the average curvature, k, of the curve at P is described as