Tractrix

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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.

Image:Tractrix.png

Mathematical derivation

Suppose the object is placed at (a,0), and the puller in the origin, so a is the length of the pulling thread. Then the puller starts to move vertically along the y axis. At every moment, the thread will be tangent to the curve y=y(x) described by the object, so it gets completely determined by the movement of the puller. Mathematically, the movement will be described then by the differential equation

<math>\frac{dy}{dx} = -\frac{\sqrt{a^2-x^2}}{x}</math>

with the initial condition y(a) = 0 whose solution is

<math>y = \int_x^a\frac{\sqrt{a^2-x^2}}{x}\,dx</math>

or

<math>y = \pm \left ( a\ln{\frac{a+\sqrt{a^2-x^2}}{x}}-\sqrt{a^2-x^2} \right ).</math>

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).

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 <math>a</math>.
• The arc length of one branch between x=x₁ and x=x₂ is <math>a \ln\left(\frac{x_1}{x_2}\right)</math>
• The area between the tractrix and its asymptote is <math>\pi a^2/2</math> 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 <math>x = a\cosh\frac{y}{a}</math>.
• The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere.

See also

Template:Commons

• 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

• Template:Planetmath reference
• Template:Planetmath reference
• Tractrix on MathWorld
• Module: Leibniz's Pocket Watch ODE at PHASERde:Traktrix

fr:tractrice ja:トラクトリックス nl:Tractrix zh:曳物线