Definition: If C be the circle of curvature at any point P on a curve, then a chord of C through P in a given direction is called the chord of curvature in that direction.
The Lengths of Chords of Curvature
Case I: Let y = ƒ(x) be any cartesian curve. Draw the circle of curvature at any point P on the curve. Let the tangent at P make and angle ? with the x-axis. Let PQ and PS be the chords of curvature parallel to x-axis and y-axis respectively. Complete the rectangle PQRS. We have PR = 2?. Now,
PQ = 2? cos (90 - ?) = 2? sin ?,
And PS = 2? cos ?.
Hence
The chord of curvature parallel to the x-axis = 2? sin ?,
The chord of curvature parallel to the y-axis = 2? cos ?.