It has been suggested that Tractrice be merged into this article or section. (Discuss)

Tractrix (from the Latin verb trahere `pull, drag') is the curve along which a small object (tractens) moves when pulled on a horizontal plane with a piece of thread by a puller (tractendus), which moves rectilinearly with infinitesimal speed. It is therefore a curve of pursuit. Image File history File links Please see the file description page for further information. ... It has been suggested that this article or section be merged into tractrix. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... Speed is the rate of motion, or equivalently the rate of change of position, many times expressed as distance d moved per unit of time t. ... A curve of pursuit is a curve constructed by analogy to having a point or points which represents pursuers and pursuees, and the curve of pursuit is the curve traced by the pursuers. ...

Tractrix with object initially at (4,0)

Image File history File links Tractrix with the parameter a=4, see Tractrix File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Tractrix with the parameter a=4, see Tractrix File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

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 An illustration of a differential equation. ...

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

or

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).

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
• The area between the tractrix and its asymptote is πa² / 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 .
• The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere.

In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ... For other uses, see Curve (disambiguation). ... In calculus, the integral of a function is an extension of the concept of a sum. ... The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). ... In geometry, a pseudosphere or tractricoid in the traditional usage, is the result of revolving a tractrix about its asymptote. ...

See also

Wikimedia Commons has media related to:

Tractrix

• Hyperbolic functions for tanh, sech, csch, arccosh
• Trigonometric function for sin, cos, tan, arccot, csc
• Signum function for sgn
• Natural logarithm for ln

Image File history File links Commons-logo. ... Wikimedia Commons logo by Reid Beels The Wikimedia Commons (also called Commons or Wikicommons) is a repository of free content images, sound and other multimedia files. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ... Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

External links

• Tractrix on PlanetMath
• Famous curves on the plane. on PlanetMath
• Tractrix on MathWorld
• Module: Leibniz's Pocket Watch ODE at PHASER