Tractrix

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Tractrix (from the Latin verb trahere "pull, drag"), or tractrice, is the curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Sir Isaac Newton (1676) and Christian Huygens(1692).

Tractrix with object initially at (4,0)
Tractrix with object initially at (4,0)

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[edit] Mathematical derivation

Suppose the object is placed at (a,0) [or (4,0) in the example shown at right], and the puller in the origin, so a is the length of the pulling thread [4 in the example at right]. Then the puller starts to move along the y axis in the positive direction. 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

\frac{dy}{dx} = -\frac{\sqrt{a^2-x^2}}{x}

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

y = \int_x^a\frac{\sqrt{a^2-t^2}}{t}\,dt = \pm \left ( a\ln{\frac{a+\sqrt{a^2-x^2}}{x}}-\sqrt{a^2-x^2} \right ).

The first term of this solution can also be written

a\ sech^{-1}\frac{x}{a},

where sech is the inverse hyperbolic secant function.

The negative branch denotes the case where the puller moves in the negative direction from the origin. Both branches belong to the tractrix, meeting at the cusp point (a, 0).

[edit] Basis of the tractrix

The essential property of the tractrix is constancy of the distance from a point P on the curve to the intersection of the y-axis and the tangent at P.

The tractrix might be regarded in a multitude of ways:

  1. It is the geometric place of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
  2. The evolvent of the function described by a fully flexible, inelastic, homogeneous string attached to two points and subjected to a gravitational field. Having the equation: y(x) = a * ch(x / a)
    note: the evolvent of the function has a perpendicular tangent to the tangent of the original function for the same x coordinate considered.
  3. The trajectory determined by the middle of the back axle of a car pulled by a rope at a constant speed and with a constant direction (initially perpendicular to the vehicle). The function admits a horizontal asymptote. The curve is symmetrical to Oy. Curvature radius r = a * ctg(x / y)

A great implication that the tractrice had was the study of the revolution surface of it around its asymptote: the pseudosphere - studied by Beltrami in 1868 with implications in interpreting the Lobachevski non-euclidian geometry.

Note: A pseudosphere has a constant negative surface, the sphere has a positive constant surface.

[edit] Properties

Catenary as evolute of a tractrix
Catenary as evolute of a tractrix
Tractrix as involute of a catenary
Tractrix as involute of a catenary
  • 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=x1 and x=x2 is a \ln\left(\frac{x_1}{x_2}\right)
  • The area between the tractrix and its asymptote is πa2 / 2 which can be found using integration or Mamikon's theorem.
  • 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}.
  • The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere.

[edit] Practical application

In 1927, P.G.A.H. Voigt patented a horn design based on the assumption that a wave front traveling through the horn is spherical of a constant radius. The idea is to minimize distortion caused by internal reflection of sound within the horn. The resulting shape is the surface of revolution of a tractrix. [1]

[edit] Drawing machines

  • In Oct.-Nov. 1692, Huygens described three tractrice drawing machines.
  • In 1693 Leibniz released to the public a machine which, in theory, could integrate any differential equation, the machine was of tractional design.
  • In 1706 John Perks built a tractional machine in order to realise the hyperbolic quadrature.
  • In 1729 Johann Poleni built a tractional device that enabled logarithmic functions to be drawn.

[edit] See also

Wikimedia Commons has media related to:

[edit] External links

[edit] Notes

  1. ^ http://www.volvotreter.de/downloads/Dinsdale_Horns_1.pdf Horn loudspeaker design pp. 4-5. (Reprinted from Wireless World, March 1974)