The Definition of Radius of Curvature

Extract

Several difficulties arise in the definition of the radius of curvature ρ of a plane curve. In the first place, it does not seem to have been previously noticed that the convention adopted in many textbooks for the sign of ρ leads to ridiculous consequences. Consider the very simple case of a circle of unit radius, with centre at the origin. Use A, B, C, D to denote the points (1, 0), (0, 1), ( −1, 0), (0, −1) respectively. Then the upper semicircle ABC is convex upwards and the lower semicircle CDA is concave upwards. G. A. Gibson, in his Elementary Treatise on the Calculus (p. 355), defines the sign of ρ at any point P as being that of d²y/dx² at that point, or, what is equivalent, as being positive or negative according as the curve is concave upwards or convex upwards at P. From this definition it follows that ρ = −1 on the upper semicircle, but ρ=+1 on the lower semicircle! At A and C d²y\dx² does not exist, so presumably ρ does not exist at these points! We cannot obtain a value by postulating continuity, since ρ is certainly discontinuous.