tractrix

A tractrix is a plane curve that arises from a simple dragging problem: a point is pulled along a straight line by a taut string of fixed length a attached to the line. If the pulling end moves along the line and the other end traces the path, the locus of the moving point is the tractrix. A key defining property is that the tangent line at any point intersects the pulling line at a fixed distance a from the point, so the segment of the tangent between the point and its intersection with the line has constant length a.

Mathematically, the tractrix can be described by the differential equation dy/dx = - y / sqrt(a^2 - y^2), with a > 0. This form reflects the constant-tangent-length condition and the asymptotic approach to the pulling line. A commonly used parametric form is x = a (t - tanh t), y = a sech t, for t ≥ 0. In this parametrization, y decreases from a to 0 while x grows without bound, and the curve is confined to the half-plane above the pulling line (with a symmetric version below for reflection).

The tractrix has a number of notable geometric features. It is asymptotic to the pulling line (the x-axis in the standard setup) and approaches it smoothly as x → ∞. The curve is convex on its standard branch and encapsulates the geometric idea of a fixed-length tether shortening the vertical component as the pulling point advances.

The name tractrix derives from Latin tractus, meaning dragging or hauling. The curve has appeared in classical studies of differential geometry and continues to be referenced in discussions of curves defined by constant tangent-length constraints and in applications involving towing paths and pursuit-type problems.