Tractrix (from the Latin verb trahere ‘pull, drag’) is the curve along which a small object (tractens) moves when pulled on a horizontal plane with a piece of thread by a puller (tractendus) which moves rectilinearly.

Let the object initially be in the xy-plane on the x-axis in the point  (a, 0)  and the puller in the origin; a is the of the pulling thread.  Then the puller begins to move along the y-axis in the positive direction.  The object follows drawing the path curve  y=y(x)  so that the line determined by the thread is at every the tangentPlanetmathPlanetmathPlanetmath of the curve.  This condition gives in the point  (x,y)  the differential equationMathworldPlanetmath


with the initial conditionMathworldPlanetmathy(a)=0.  The solution is




Here the minus alternative is for the case that the puller moves in the negative direction from the origin.  In fact, both branches, corresponding to both signs, belong to the tractrix.  The branches meet in the cusp point  (a, 0).

The substitution  x:=acost  gives for the tractrix the parametric


Another one is


where cosh and tanh are the hyperbolic functionsDlmfMathworldPlanetmath cosinus hyperbolicus and tangens hyperbolica.


  1. 1.

    It is obvious that the line, on which the puller goes, is the asymptote of the tractrix.  The curve thus has the property that its tangent, between the asymptote and the point of tangency, has the (=a).

  2. 2.

    The differential equation of the orthogonal curves of the tractrix is


    whence they are the circles  x2+(y-C)2=a2.

  3. 3.

    The arc lengthMathworldPlanetmath of one branch on the interval[b,a] is simply

  4. 4.

    The area A between the tractrix and its asymptote is πa22.   This may be calculated ordinarily as


    integrating by parts and using the area of a quarter-circle yield


    and moreover


    (see this entry ( for  limx0+xlnx=0).  Another way to determine A is differential-geometric:  as the object draws the tractrix from above to down, the thread turns 180o and thus sweeps an area equal to a half-circle.

  5. 5.

    The envelope of the normal lines of the tractrix, i.e. the evolute of the tractrix is the catenary (or “chain curve”)  x=acoshya.

Title tractrix
Canonical name Tractrix
Date of creation 2013-03-22 15:18:32
Last modified on 2013-03-22 15:18:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 26
Author pahio (2872)
Entry type Derivation
Classification msc 51N05
Related topic SubstitutionNotation
Related topic Catenary
Related topic EulersSubstitutionsForIntegration
Defines tractrix