This means that The evolute of a curve K is the locus of the center of the osculating circles and the envelope of all osculating circles is K itself. The unknowns are a, b and r. The inverse of this curvature is called the radius of curvature. This is the same formula as for the radius of the osculating circle. The catenary is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity.
All catenaries are homothetic transformations of the graph of the cosh function.
Now we'll calculate the parametric equations of the locus of the intersection points of the tangent line in P and the line through S, perpendicular to that tangent line. If we solve the system of these two equations for x and y, it will give the parametric equations of the locus. This locus is called the tractrix associated to the catenary. From this, we see that the tangent line in T to the tractrix is orthogonal to the tangent line in P to the catenary . The normal in T to the tractrix is tangent to the catenary.