Tractrix

A tractrix (from the Latin verb trahere "pull, drag"; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1692).

Suppose the object is placed at (a,0) (or (4,0) in the example shown at right}, and the puller in the origin, so a is the length of the pulling thread (4 in the example at right). Then the puller starts to move along the y axis in the positive direction. At every moment, the thread will be tangent to the curve y = y(x) described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, the movement will be described then by the differential equation

with the initial condition y(a) = 0 whose solution is

The first term of this solution can also be written

where arsech is the inverse hyperbolic secant function.

The negative branch denotes the case where the puller moves in the negative direction from the origin. Both branches belong to the tractrix, meeting at the cusp point (a,0).

The essential property of the tractrix is constancy of the distance between a point P on the curve and the intersection of the tangent line at P with the asymptote of the curve.

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