tractrix

The study of this curve started with a problem stated by Huygens (1692): given a inextensible string, moving in a straight line, dragging an object. What then, is the path of that object, asked Huygens himself. And he gave the curve the name of the tractrix. Also Leibniz and Johann Bernoulli studied the curve.

Because the dragging quality the curve has been given the names drag curve¹⁾ and tractory curve. Also the name of donkey curve²⁾ can be heard.
For the tractrix the length of a tangent from the tangent point to the asymptote is constant, leading to its alternative name of equitangential curve.

The curve must not be confused with the pursuit curve, while there is no fixed length between dragging and dragged object.

Relationships with other curves are the following:

• its evolute is the catenary
• its radial is the kappa curve
• when a hyperbolic spiral is rolling over a line, then the path of the pole is a tractrix

Beltrami (1868) used the tractrix to rotate around its asymptote, in his study of non-Euclidean geometry. The surface of the resulting pseudosphere has a constant negative curvature.

notes

1) trahere (Lat.) = to drag
In German: Schleppkurve.

2) In French: courbe de l'âne