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

The tractrice is a curve, first introduced by Claude Perrault in 1670, and later studied by Sir Isaac Newton (1676) and Christian Huygens(1692). Wikipedia does not have an article with this exact name. ... It has been suggested that Tractrice be merged into this article or section. ... Though Claude Perrault (Paris, 1613 - Paris, 1688) is best known as the architect of the eastern range of the Louvre in Paris, he also achieved success as physician and anatomist, and as an author, who wrote treatises on physics and natural history. ... Sir Isaac Newton in Knellers portrait of 1689. ... Christiaan Huygens Christiaan Huygens (approximate pronunciation: HOW-khens; SAMPA /h9yGEns/ or /h@YG@ns/) (April 14, 1629–July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ...

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.

Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... For hyperbole, the figure of speech, see hyperbole. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Partial plot of a function f. ...

Basis of the tractrice

The essential property of the tractrice is that the length of the tangent] to it and the x axis remains constant at any given point. In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... A coordinate axis is one of a set of vectors that defines a coordinate system. ...

It might be regarded in a multitude of ways:

1. It is the geometric place of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
2. 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 * ch(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.
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). The function admits a horizontal asymptote. The curve is symmetrical to Oy. Curvature radius r = a * ctg(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. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... The term elastomer is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. ... Beltrami refers to more than one thing: Eugenio Beltrami, a mathematician Beltrami County, Minnesota The Minnesota city of Beltrami A neighborhood in Minneapolis, Minnesota (Beltrami, Minneapolis) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...

Note: A pseudosphere has a constant negative surface, the sphere has a positive constant surface.

The tractrice equation

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

1. (trigonometric) :
   x = a * [argch(a / x) − (a² − y²)⁽1 / 2)]
   x = a * ln[(a + (a² − y²)⁽1 / 2)] / y − (a² − y²)⁽1 / 2)
   y = a * cos(t) where t belongs to [0;pi/2]
2. (hyperbolic) :
   y = a / cosh(t)
3. (differential) :
   dx / xy = − [y / (a² − y²)⁽1 / 2)]