Tractrix
A plane transcendental curve whose equation in rectangular Cartesian coordinates has the form
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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:
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The radius of curvature is:
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The area bounded by the tractrix and its asymptote is:
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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.
References
[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |
Comments
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) |
Tractrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tractrix&oldid=49006