Tractrix

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

$$ x = \pm a \mathop{\rm ln} \ \frac{a + \sqrt {a ^ {2} - y ^ {2} } }{y } \mp \sqrt {a ^ {2} - y ^ {2} } . $$

The tractrix is symmetric about the origin (see Fig.), the $ x $- axis being an asymptote. The point $ ( 0, a) $ is a cusp with vertical tangent. The length of the arc measured from the point $ x = 0 $ is:

$$ l = a \mathop{\rm ln} { \frac{a}{y} } . $$

The radius of curvature is:

$$ r = a \mathop{\rm cot} { \frac{x}{y} } . $$

The area bounded by the tractrix and its asymptote is:

$$ S = { \frac{\pi a ^ {2} }{2} } . $$

Figure: t093570a

The rotation of the tractrix around the $ x $- axis generates a pseudo-sphere. The length of the tangent, that is, of the segment between the point of tangency $ M $ and the $ x $- axis, is constant. This property enables one to regard the tractrix as the trajectory of the end of a line segment of length $ a $, when the other end moves along the $ x $- 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.

References

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

References

[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)