Radius of Curvature of Catenary
Consider the catenary (blue curve). Let be an arbitrary point on the catenary and let be the point where the normal to the catenary meets the axis. Then the radius of curvature of the catenary at is equal to the distance from to , that is, , where is the center of the osculating circle to the catenary at .
 V. P. Minorsky, Problems in Higher Mathematics, Moscow: Mir Publishers, 1975, p. 217.