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A plane transcendental curve whose equation in rectangular Cartesian coordinates has the form

The tractrix is symmetric about the origin (see Fig.), the -axis being an asymptote. The point is a cusp with vertical tangent. The length of the arc measured from the point is:

The radius of curvature is:

The area bounded by the tractrix and its asymptote is:

Figure: t093570a

The rotation of the tractrix around the -axis generates a pseudo-sphere. The length of the tangent, that is, of the segment between the point of tangency and the -axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length , when the other end moves along the -axis. The notion of a tractrix generalizes to the case when the end of the segment does not move along a straight line, but along some given curve; the curve obtained in this way is called the trajectory of the given curve.


[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)



[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
[a3] M. Greenberg, "Euclidean and non-Euclidean geometries" , Freeman (1974)
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish pp. 1–5
[a5] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)
[a6] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Tractrix. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098