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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.



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

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.

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.