View source for Tractrix

{{short description|Curve traced by a point on a rod as one end is dragged along a line}}
[[Image:Tractrixtry.gif|thumb|500px|Tractrix created by the end of a pole (lying flat on the ground). Its other end is first pushed then dragged by a finger as it spins out to one side.]]

In [[geometry]], a '''tractrix''' ({{ety|la|trahere|to pull, drag}}; 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).<ref>{{cite book |title=Mathematics and Its History |edition=revised, 3rd |first1=John |last1=Stillwell |publisher=Springer Science & Business Media |year=2010 |isbn=978-1-4419-6052-8 |page=345 |url=https://books.google.com/books?id=V7mxZqjs5yUC}}, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA345 extract of page 345]</ref>

==Mathematical derivation==
[[Image:Tractrix.svg|thumb|180px|Tractrix with object initially at {{math|(4, 0)}}]]
Suppose the object is placed at {{math|(''a'', 0)}} and the puller at the [[origin (mathematics)|origin]], so that {{mvar|a}} is the length of the pulling thread. (In the example shown to the right, the value of {{math|''a''}} is 4.) Suppose the puller starts to move along the {{mvar|y}} axis in the positive direction. At every moment, the thread will be [[tangent]] to the curve described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, if the coordinates of the object are {{math|(''x'', ''y'')}}, then by the [[Pythagorean theorem]] the {{nowrap|{{mvar|y}}-coordinate}} of the puller is <math>y + \sqrt{a^2 - x^2}</math> . Writing that the slope of thread equals that of the tangent to the curve leads to the [[differential equation]]
:<math>\frac{dy}{dx} = -\frac{\sqrt{a^2-x^2}}{x}</math>
with the initial condition {{math|1=''y''(''a'') = 0}}. Its solution is
:<math>y = \int_x^a \frac{\sqrt{a^2-t^2}}{t}\,dt = \!  a\ln{\frac{a+\sqrt{a^2-x^2}}{x}}-\sqrt{a^2-x^2} .</math>
If instead the puller moves downward from the origin, then the sign should be removed from the
differential equation and therefore inserted into the solution. Each of the two solutions defines a branch of the tractrix; they meet at the [[cusp (singularity)|cusp]] point {{math|(''a'', 0)}}.

The first term of this solution can also be written 
:<math>a \operatorname{arsech}\frac{x}{a}, </math> 
where {{math|arsech}} is the [[inverse hyperbolic secant]] function.



==Basis of the tractrix==
The essential property of the tractrix is constancy of the distance between a point {{mvar|P}} on the curve and the intersection of the [[tangent line]] at {{mvar|P}} with the [[asymptote]] of the curve.
 
The tractrix might be regarded in a multitude of ways:
 
# It is the [[locus (mathematics)|locus]] of the center of a [[hyperbolic spiral]] rolling (without skidding) on a straight line.
# It is the [[involute]] of the [[catenary]] function, which describes a fully flexible, [[elastomer|inelastic]], homogeneous string attached to two points that is subjected to a [[gravity|gravitational]] field. The catenary has the equation {{math|1=''y''(''x'') = ''a'' cosh&thinsp;{{sfrac|''x''|''a''}}}}.
#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).
# It is a (non-linear) curve which a [[circle]] of radius {{math|''a''}} rolling on a straight line, with its center at the {{math|''x''}} axis, intersects perpendicularly at all times.

The function admits a horizontal asymptote. The curve is symmetrical with respect to the {{mvar|y}}-axis. The curvature radius is {{math|1=''r'' = ''a'' cot&thinsp;{{sfrac|''x''|''y''}}}}.

A great implication that the tractrix had was the study of its [[surface of revolution]] about its asymptote: the [[pseudosphere]]. Studied by [[Eugenio Beltrami]] in 1868,<ref>{{cite journal
 | last = Beltrami | first = E.
 | journal = Giornale di Matematiche
 | page = 284
 | title = Saggio di interpretazione della geometria non euclidea
 | volume = 6
 | year = 1868}} As cited by {{cite book
 | last1 = Bertotti | first1 = Bruno
 | last2 = Catenacci | first2 = Roberto
 | last3 = Dappiaggi | first3 = Claudio
 | arxiv = math/0506395
 | contribution = Pseudospheres in geometry and physics: from Beltrami to de Sitter and beyond
 | isbn = 978-88-7916-359-0
 | mr = 2374676
 | pages = 165–194
 | publisher = LED–Ed. Univ. Lett. Econ. Diritto, Milan
 | series = Ist. Lombardo Accad. Sci. Lett. Incontr. Studio
 | title = A great mathematician of the nineteenth century. Papers in honor of Eugenio Beltrami (1835–1900) (Italian)
 | volume = 39
 | year = 2007}}</ref> as a surface of constant negative [[Gaussian curvature]], the pseudosphere is a local model of [[hyperbolic 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.<ref>{{cite book|title=Mathematics and the Imagination|title-link=Mathematics and the Imagination|series=Dover Books on Mathematics|first1=Edward|last1=Kasner|first2=James|last2=Newman|publisher=Courier Corporation|year=2013|isbn=9780486320274|contribution=Figure 45(a)|page=141|contribution-url=https://books.google.com/books?id=-bXDAgAAQBAJ&pg=PA141}}</ref>

==Properties==
[[Image:Evolute2.gif|thumb|500px|right|[[Catenary]] as [[evolute]] of a tractrix]]
<!-- [[Image:Involute.gif|thumb|500px|right|Tractrix as [[evolute]] of a catenary]] -->
*The curve can be parameterised  by the equation <math>x = t - \tanh(t), y= 1/{\cosh(t)}</math>.<ref>{{MacTutor|class=Curves|id=Tractrix|title=Tractrix}}</ref>
* 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 {{mvar|a}}.
* The [[arc length]] of one branch between {{math|1=''x'' = ''x''<sub>1</sub>}} and {{math|1=''x'' = ''x''<sub>2</sub>}} is {{math|''a''&thinsp;ln&thinsp;{{sfrac|''y''<sub>1</sub>|''y''<sub>2</sub>}}}}.
* The [[area]] between the tractrix and its asymptote is {{math|{{sfrac|π&hairsp;''a''<sup>2</sup>|2}}}}, which can be found using [[integral|integration]] or [[Mamikon's theorem]].
* The [[envelope (mathematics)|envelope]] of the [[surface normal|normal]]s of the tractrix (that is, the [[evolute]] of the tractrix) is the [[catenary]] (or ''chain curve'') given by {{math|1=''y'' = ''a'' cosh&thinsp;{{sfrac|''x''|''a''}}}}.
* 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.

==Practical application==
In 1927, P.&nbsp;G.&nbsp;A.&nbsp;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#Audio_distortion|distortion]] caused by internal reflection of sound within the horn. The resulting shape is the surface of revolution of a tractrix.<ref>[http://www.volvotreter.de/downloads/Dinsdale_Horns_1.pdf Horn loudspeaker design pp. 4–5. (Reprinted from Wireless World, March 1974)]</ref> Voigt's design removed the annoying "honk" characteristic from previous horn designs, especially conical horns, and thus revitalized interest in the horn loudspeaker.<ref>{{cite book |last=Self |first=Douglas |date=2012 |title=Audio Engineering Explained |page=334 |publisher=Taylor & Francis |isbn=9781136121258}}</ref> [[Klipsch Audio Technologies]] has used the tractrix design for the great majority of their loudspeakers, and many loudspeaker designers returned to the tractrix in the 21st century, creating an [[audiophile]] market segment. The tractrix horn differs from the more common exponential horn in that it provides for a wider spread of high frequency energy, and it supports the lower frequencies more strongly.<ref>{{cite book |last=Eargle |first=John |date=2013 |title=Loudspeaker Handbook |publisher=Springer Science |page=164 |isbn=9781475756784}}</ref>

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.<ref>{{Cite book|title = Handbook of Metal Forming|last = Lange|first = Kurt|publisher = McGraw Hill Book Company|year = 1985|pages = 20.43}}</ref>

A [[toothed belt]]-pulley design provides improved efficiency for mechanical power transmission using a tractrix catenary shape for its teeth.<ref>{{cite web|url=https://www.gates.com/~/media/files/gates/industrial/power-transmission/manuals/powergripdrivedesignmanual_17195_2014.pdf|title=Gates Powergrip GT3 Drive Design Manual|date=2014|access-date=17 November 2017|website=Gates Corporation|quote=The GT tooth profile is based on the tractix mathematical function. Engineering handbooks describe this function as a “frictionless” system. This early development by Schiele is described as an involute form of a catenary.|page=177}}</ref>  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 [[trapezoid]]al or circular tooth shapes, which cause significant sliding and friction.

==Drawing machines==
* In October–November 1692, Christiaan Huygens described three tractrix-drawing machines.<ref name="bos"/>
* In 1693 [[Gottfried Wilhelm Leibniz]] devised a "universal tractional machine" which, in theory, could integrate any [[Ordinary differential equation#Definitions|first order differential equation]].<ref>{{cite book|title=From Logic to Practice: Italian Studies in the Philosophy of Mathematics|quote=... mechanical devices studied ... to solve particular differential equations ... We must recollect Leibniz's 'universal tractional machine'|first=Pietro|last=Milici|editor-first=Gabriele|editor-last=Lolli|publisher=Springer|date=2014}}</ref>  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 function|hyperbolic]] quadrature.<ref>{{cite journal|last1=Perks|first1=John|title=The construction and properties of a new quadratrix to the hyperbola|journal=Philosophical Transactions|date=1706|volume=25|pages=2253–2262|jstor=102681|doi=10.1098/rstl.1706.0017|s2cid=186211499 }}</ref>
* In 1729 [[Giovanni Poleni]] built a tractional device that enabled [[logarithmic function]]s to be drawn.<ref>{{cite book|last1=Poleni|first1=John|title=Epistolarum mathematicanim fasciculus|date=1729|page=letter no. 7}}</ref>
A history of all these machines can be seen in an article by [[H. J. M. Bos]].<ref name="bos">{{cite journal|last1=Bos|first1=H.&nbsp;J.&nbsp;M.|title=Recognition and Wonder – Huygens, Tractional Motion and Some Thoughts on the History of Mathematics|url=http://www.gewina.nl/journals/tractrix/bos89.pdf|journal= Euclides|volume= 63 |date=1989 |pages=65–76}}</ref>

==See also==
*[[Dini's surface]]
*[[Hyperbolic functions]] for {{math|tanh}}, {{math|sech}}, {{math|csch}}, {{math|arcosh}}
*[[Natural logarithm]] for {{math|ln}}
*[[Sign function]] for {{math|sgn}}
*[[Trigonometric functions]] for {{math|sin}}, {{math|cos}}, {{math|tan}}, {{math|arccot}}, {{math|csc}}

==Notes==
{{reflist}}

==References==
* {{cite book|first1=Edward|last1=Kasner|first2=James|last2=Newman|date=1940|title=Mathematics and the Imagination|page=[https://archive.org/details/mathematicsimagi00kasnrich/page/141 141–143]|publisher=[[Simon & Schuster]]|title-link=Mathematics and the Imagination}}
* {{cite book | first=J. Dennis | last=Lawrence | title=A Catalog of Special Plane Curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/5 5, 199] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/5 }}

==External links==
{{commons|Tractrix}}
* {{MacTutor|class=Curves|id=Tractrix|title=Tractrix}}
* {{planetmath reference|urlname=Tractrix|title=Tractrix}}
* {{planetmath reference|urlname=FamousCurves|title=Famous curves}}
*[http://mathworld.wolfram.com/Tractrix.html Tractrix] on [[MathWorld]]
*[http://www.phaser.com/modules/historic/leibniz/ Module: Leibniz's Pocket Watch ODE] at PHASER

[[Category:Plane curves]]
[[Category:Mathematical physics]]