Curvature

One of the typical application of derivative is curvature. By definition, the simplest form of curvature k at a given point on a curve is the rate of change of the tangential angle ΔΨ between tangents with respect to the arc length Δs.

 

Curvature 

Mathematically:

 

Assume the radius of circle at the given point is r. Geometrically:

 

This implies the curvature of a curve at a given point equal to 1/radius. The radius is an intantaneous radius of the curve at the given point. If the curve is a circle, the curvature at all point is the same and is equal to 1/radius of the circle.

Radius of Curvature 

Therefore, a circle of the same curvature can be constructed at any point along the curve and this circle is known as the circle of curvature of the curve at the given point. The radius of the circle at the given point is always normal to the curve. This radius is called the radius of curvature of the curve at the given point and is denoted by ρ. Imply

 

Since both Gs and GZ are functions of x and as Gx approaching 0, both Gs and GZ approaching 0 also, imply the radius of curvature can be expressed as,

 

For the arc of ΔΨ ,as Δx approaching 0, arc of ΔΨ can be approximated by chord of ΔΨ ,

Applying Pythagorean theorem, imply

 

As Δx approaching 0, take limit.

 

For the angle ΔΨ, the angle can be expressed as the slope of the given point at (x,y), tan Ψ and is equal to the derivative, dy/dx of the function at the given point, imply:

 

Differentiating the equation with respect to x to get the dΨ/dx, imply

Therefore the radius of curvature in terms of x and y is

And the curvature in terms of x and y is