Tractrix

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, it is therefore a curve of pursuit.

Tractrix with object initially at (4,0)

Properties

• Due to the geometrical way it was defined, the tractrix has the property that the length of its tangent, between the asymptote and the point of tangency, has constant length $a$.
• The arc length of one branch between x=a and x=b is $a\ln \left({\frac {a}{b}}\right)$
• The area between the tractrix and its asymptote is $\pi a^{2}/2$ which can be found using integration.
• The envelope of the normals of the tractrix, that is, the evolute of the tractrix is the catenary (or chain curve) given by $x=a\cosh {\frac {y}{a}}$.

See also

• Hyperbolic functions for tanh, sech, csch, arccosh
• Trigonometric function for sin, cos, tan, arccot, csc
• Signum function for sgn
• Natural logarithm for ln

External links

• "Tractrix". PlanetMath.
• "Famous curves on the plane". PlanetMath.
• Tractrix on MathWorld
• Module: Leibniz's Pocket Watch ODE at PHASER