Трактрикс қисығы: геометриялық анықтамасы және қолданыстары

Curve traced by a point on a rod as one end is dragged along a line

In geometry, a tractrix (; plural: tractrices) 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 pulling point (the tractor) 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 Isaac Newton (1676) and Christiaan Huygens (1693).

The curve can be parameterised by the equation Due to the geometrical way it was defined, the tractrix has the property that the segment of its tangent, between the asymptote and the point of tangency, has constant length a. The arc length of one branch between 1=x = x1 and 1=x = x2 is The area between the tractrix and its asymptote is , 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 The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere. The tractrix is a transcendental curve; it cannot be defined by a polynomial equation.

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. An important application is in the forming technology for sheet metal. In particular a tractrix profile is used for the corner of the die on which the sheet metal is bent during deep drawing. A toothed belt pulley design provides improved efficiency for mechanical power transmission using a tractrix catenary shape for its teeth. This shape minimizes the friction of the belt teeth engaging the pulley, because the moving teeth engage and disengage with minimal sliding contact. Original timing belt designs used simpler trapezoidal or circular tooth shapes, which cause significant sliding and friction.

In October–November 1692, Christiaan Huygens described three tractrix drawing machines. The concept was an analog computing mechanism implementing the tractional principle. The device was impractical to build with the technology of Leibniz's time, and was never realized. In 1706 John Perks built a tractional machine in order to realise the hyperbolic quadrature. In 1729 Giovanni Poleni built a tractional device that enabled logarithmic functions to be drawn. A history of all these machines can be seen in an article by H. J. M. Bos.