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Tractrix

The tractrix is a curve used to describe mathematical, physical and engineering situations and proceses.

Author: Dragan Valeriu

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The tractrix (lat. trahere~to pull) is a curve, first introduced by Claude Perrault in 1670 and later on studied by sir Isaac Newton-1676 and Christian Huygens-1692.

In oct.-nov. 1692, Huygens describes three tractrix drawing machines.
In 1693 Leibniz releases to the public a machine wich, in theory, could integrate any differential equation, the machine was of tractional design.
In 1706 John Perks builds a tractional machine in order to realise the hyperbolic cuadrature
In 1729 Johann Poleni builds a tractional device that enabled logarithmic functions to be drawn.

The essential property of the tractrix is thet the lenght of the tangent to it and the Ox axis remains constant at any given point.

It might be regarded in a multitude of ways:

1. It is the geometric place of the center of an hyperbolic spiral rolling (witout skidding) on a streight line.

2. The evolvent of the function described by a fully flexible, non-elastic, homogenous string atached 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 axel of a car pulled by a rope at a constant speed and with a constant direction (initialy perpendicular to the vehicle)

   The function admites an horisontal asymptote
   The curve is symetrical 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 it-s 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.

The tractrix equation :

the coordinates of the turning point A(x;y)=(0;a)

1.(trigonometric):
   x=a*[argch(a/x)-(a^2 -y^2)^(1/2)]
   x=a*ln{[(a+ (a^2 - y^2)^(1/2)]/y}-(a^2 - y^2)^(1/2)
   y=a* cos(t) where t belongs to [0;pi/2]

2. (hyperbolic) :
y = a/cosh(t)

3. (differential):
dx/xy =- [y/(a^2 - y^2)^(1/2)]

About Author

Aerospace engeneering student at Universitatea Politehnica Bucuresti

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