Chord of Curvature:

Let C be the circle of curvature at any arbitrary point P on a curve, then a chord of C through P in a provided direction is known as the chord of curvature in that direction.

The Lengths of Chords of Curvature

Case I: Let y = ƒ(x) be Cartesian curve. Draw the circle of curvature at point P on the curve. Let the tangent at P create and angle ψ with the x-axis. Let PS and PQ be the chords of curvature parallel to y-axis and x-axis respectively. Complete the rectangle PQRS. We have PR = 2ρ. Now,

PQ = 2ρ cos (90 - ψ) = 2ρ sin ψ,

And PS = 2ρ cos ψ.

Therefore

The chord of curvature parallel to the x-axis = 2ρ sin ψ,

The chord of curvature parallel to the y-axis = 2ρ cos ψ.

Case II: Let r = ƒ(θ) be a polar curve. Let PN and PL be the chords of curvature through the pole and perpendicular to the radius vector OP respectively. We have PM = 2ρ.

Now PL = 2ρ cos (π/2- Ø) = 2ρ sin Ø,

And, PN = 2ρ cos Ø.

Therefore,

The chord of curvature through the pole = 2ρ sin Ø,

The chord of curvature perpendicular to the radius vector = 2ρ cos Ø.