Note on the Determination of Centres of Curvature

For determining the Cartesian co-ordinates of the centre of curvature of a plane curve two methods are principally used in the text-books. One of these (see for example, Edwards, “Differential Calculus,” p 266, § 339) having previously established the formula for the radius of curvature, derives the coordinates of the centre of curvature by using the circular functions of the angle “ψ,” which the tangent to the curve at the point considered makes with OX. Since, however, for the same tangent, and therefore the same “ψ” the curve may be either convex or concave towards OX (and accordingly the centre of curvature have quite different positions), this method introduces an essential ambiguity, and is not to be recommended.