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


One parametric form is


The arc length, curvature, and tangential angle for this parameterization with a=1 are


where gd(t) is the Gudermannian.

Rather surprisingly, area under the curve is given by


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


then solving for t and plugging back in to obtain


(Gray 1997). This parameterization has curvature


In terms of the angle theta^'=pi/2+theta, the parametric equations can be written


(Lockwood 1967, p. 123), where gd^(-1)x is the inverse Gudermannian.

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

x(t)={ae^(-v/a) for v in [0,infty); ae^(v/a) for v in (-infty,0]
y(t)={a[tanh^(-1)(sqrt(1-e^(-2v/a)))-sqrt(1-e^(-2v/a))] for v in [0,infty); a[-tanh^(-1)(sqrt(1-e^(2v/a)))+sqrt(1-e^(2v/a))] for v in (-infty,0].

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.

See also

Curvature, Dini's Surface, Gudermannian, Mice Problem, Pseudosphere, Pursuit Curve, Tractrix Spiral

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Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 226, 1987.Geometry Center. "The Tractrix.", A. "The Tractrix" and "The Evolute of a Tractrix is a Catenary." §3.6 and 5.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 61-64 and 102-103, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 199-200, 1972.Lockwood, E. H. "The Tractrix and Catenary." Ch. 13 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 118-124, 1967.MacTutor History of Mathematics Archive. "Tractrix.", D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 330, 1958.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 249-251, 1999.Yates, R. C. "Tractrix." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 221-224, 1952.

Cite this as:

Weisstein, Eric W. "Tractrix." From MathWorld--A Wolfram Web Resource.

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