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曳物线 （英文：tractrix）是

is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed.因此它是一条追逐曲线。Claude Perrault在1670年首次引入此概念，后来牛顿 (1676)和惠更斯 (1692)也曾研究过它。

Tractrix with object initially at (4,0)

数学推导

假设物体被放置在点(a,0) [右图中是(4,0)]，而牵引物在原点,那么a就是绳子的长度[图中为4]。然后牵引物开始沿y轴正方向移动。在每个瞬间，绳子都相切于物体画出的曲线$y=y(x)$，于是该曲线可由以下的微分方程描述：

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

加上初值条件y(a) = 0，可解出

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

第一项也可以写成：

$a\ \mathrm {arsech} {\frac {x}{a}},\,\!$

其中arsech是反双曲正割函数。

带负号的一支对应于牵引物从原点沿y轴负方向运动的情形。这两支都属于曳物线，它们相交于尖点(a, 0)。

Basis of the tractrix

The essential property of the tractrix is constancy of the distance between a point P on the curve and the intersection of the tangent line at P with the asymptote of the curve.

The tractrix might be regarded in a multitude of ways:

1. It is the locus of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
2. The evolute of the catenary function, which describes a fully flexible, inelastic, homogeneous string attached to two points that is subjected to a gravitational field. The catenary has the equation $y(x)=a\,\operatorname {cosh} (x/a)$.
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).

Tractrix by dragging a pole.

The function admits a horizontal asymptote. The curve is symmetrical with respect to the y-axis. The curvature radius is $r=a\,\operatorname {cot} (x/y)$

A great implication that the tractrix had was the study of the revolution surface of it around its asymptote: the pseudosphere. Studied by Beltrami in 1868, as a surface of constant negative Gaussian curvature, the pseudosphere is a local model of non-Euclidean geometry. The idea was carried further by Kasner and Newman in their book Mathematics and the Imagination, where they show a toy train dragging a pocket watch to generate the tractrix.

Properties

Catenary as involute of a tractrix

Practical application

In 1927, P.G.A.H. Voigt patented a horn loudspeaker 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.
Tractrices are also the paths that missiles and torpedoes take as they approach their respective targets.

Drawing machines

• In October–November 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.

See also

• Dini's surface
• Hyperbolic functions for tanh, sech, csch, arccosh
• Natural logarithm for ln
• Sign function for sgn
• Trigonometric function for sin, cos, tan, arccot, csc

References

• Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 141–3, Simon & Schuster.
• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 5, 199. ISBN 0-486-60288-5.

External links

• O'Connor, John J.; Robertson, Edmund F., "Tractrix", MacTutor History of Mathematics Archive, University of St Andrews
• "Tractrix". PlanetMath.
• "Famous curves on the plane". PlanetMath.
• Tractrix on MathWorld
• Module: Leibniz's Pocket Watch ODE at PHASER