Tractrix

At every moment, the thread will be tangent to the curve described by the object, so that it becomes completely determined by the movement of the puller.

Writing that the slope of thread equals that of the tangent to the curve leads to the differential equation with the initial condition y(a) = 0.

The first term of this solution can also be written where arsech is the inverse hyperbolic secant function.

Studied by Eugenio Beltrami in 1868,[2] as a surface of constant negative Gaussian curvature, the pseudosphere is a local model of hyperbolic geometry.

The idea was carried further by Kasner and Newman in their book Mathematics and the Imagination, where they show a toy train dragging a pocket watch to generate the tractrix.

In particular a tractrix profile is used for the corner of the die on which the sheet metal is bent during deep drawing.

[8] A toothed belt-pulley design provides improved efficiency for mechanical power transmission using a tractrix catenary shape for its teeth.

Original timing belt designs used simpler trapezoidal or circular tooth shapes, which cause significant sliding and friction.