Tractrix

The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog's master, the curve has the descriptive name "hundkurve" (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).

From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve.

In Cartesian coordinates, the tractrix has equation

(1)

One parametric form is

			(2)
			(3)

The arc length, curvature, and tangential angle for this parameterization with are

			(4)
			(5)
			(6)

where is the Gudermannian.

Rather surprisingly, area under the curve is given by

(7)

A second parametric form in terms of the angle of the straight line tangent to the tractrix can be found by computing

			(8)
			(9)
			(10)

then solving for and plugging back in to obtain

			(11)
			(12)
			(13)

(Gray 1997). This parameterization has curvature

(14)

In terms of the angle , the parametric equations can be written

			(15)
			(16)
			(17)
			(18)

(Lockwood 1967, p. 123), where is the inverse Gudermannian.

A parameterization which traverses the tractrix with constant speed is given by

			(19)
			(20)

When a tractrix is rotated around its asymptote, a pseudosphere results. This is a surface of constant negative curvature. For a tractrix, the length of a tangent from its point of contact to an asymptote is constant. The area between the tractrix and its asymptote is finite.