Tractrix

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Tractrix (from the Latin verb trahere "pull, drag"), or tractrice, is the curve along which a small object moves when pulled on a horizontal plane with a piece of thread by a puller, which moves rectilinearly with 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)

1 Drawing machines
2 Mathematical derivation
3 The tractrix equation
4 Basis of the tractrix
5 Properties
6 See also
7 External links

[edit] Drawing machines

In Oct.-Nov. 1692, Huygens describes three tractrice drawing machines.
In 1693 Leibniz releases to the public a machine which, 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 quadrature.
In 1729 Johann Poleni builds a tractional device that enabled logarithmic functions to be drawn.

[edit] 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

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

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

$y = \int_x^a\frac{\sqrt{a^2-x^2}}{x}\,dx$

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

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).

[edit] The tractrix equation

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

(trigonometric) :
$x = a * [a r g c h (a / x) - (a 2 - y 2) (1 / 2)]$
$x = a * l n [(a + (a 2 - y 2) (1 / 2)] / y - (a 2 - y 2) (1 / 2)$
$y = a * c o s (t)$ where t belongs to [0;pi/2]
(hyperbolic) :
$y = a / c o s h (t)$
(differential) :
$d x / x y = - [y / (a 2 - y 2) (1 / 2)]$

[edit] Basis of the tractrix

The essential property of the tractrix is that the length of the tangent to it and the x axis remains constant at any given point.

It might be regarded in a multitude of ways:

It is the geometric place of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
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: $y (x) = a * c h (x / a)$
note: the evolvent of the function has a perpendicular tangent to the tangent of the original function for the same x coordinate considered.
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 $r = a * c t g (x / y)$

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

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 $π a 2 / 2$ 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 $x = a\cosh\frac{y}{a}$ .
The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere.