|
|
| Line 1: |
Line 1: |
| '''Tractrix''' (from the [[Latin]] verb ''trahere'' "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 tractor (pulling) point 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 [[Sir Isaac Newton]] (1676) and [[Christiaan Huygens]] (1692).
| | Irwin Butts is what my spouse loves to call me although [http://Brown.edu/Student_Services/Health_Services/Health_Education/sexual_health/sexually_transmitted_infections/hpv.php I don't] really like being called like that. North Dakota is her beginning place but she will have to move one at home std testing working day or another. Hiring has been my occupation for some time but I've currently applied for an additional one. home std test kit To [http://Www.Myelectronicmd.com/get_reference.php?Id=1296&condition=ANAL%20HERPES%20SIMPLEX%20VIRUS&symname=A&typ=3 play baseball] is the hobby he will by no means quit doing.<br><br>Review my homepage - [http://www.streaming.iwarrior.net/user/WMcneil www.streaming.iwarrior.net] |
| | |
| [[Image:Tractrix.png|thumb|180px|right|Tractrix with object initially at (4,0)]]
| |
| | |
| ==Mathematical derivation==
| |
| Suppose the object is placed at (''a'',0) [or (4,0) in the example shown at right], and the puller in the [[origin (mathematics)|origin]], so ''a'' is the length of the pulling thread [4 in the example at right]. Then the puller starts to move along the ''y'' axis in the positive direction. At every moment, the thread will be tangent to the curve ''y'' = ''y''(''x'') described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, the movement will be described then by the [[differential equation]]
| |
| :<math>\frac{dy}{dx} = -\frac{\sqrt{a^2-x^2}}{x}\,\!</math>
| |
| with the initial condition ''y(a)'' = 0 whose solution is
| |
| :<math>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 ).\,\!</math>
| |
| | |
| The first term of this solution can also be written
| |
| :<math>a\ \mathrm{arsech}\frac{x}{a}, \,\!</math>
| |
| where ''arsech'' is the [[inverse hyperbolic secant]] function.
| |
| | |
| The negative branch denotes the case where the puller moves in the negative direction from the origin. Both branches belong to the tractrix, meeting at the [[cusp (singularity)|cusp]] point (''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:
| |
|
| |
| # It is the [[locus (mathematics)|locus]] of the center of a hyperbolic spiral rolling (without skidding) on a straight line.
| |
| # The [[involute]] of the [[catenary]] function, which describes a fully flexible, [[elastomer|inelastic]], homogeneous string attached to two points that is subjected to a gravitational field. The catenary has the equation <math>y(x)=a\,\operatorname{cosh}(x/a)</math>.
| |
| #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).
| |
| [[Image:Tractrixtry.gif|thumb|500px|right|Tractrix by dragging a pole.]]
| |
| The function admits a horizontal asymptote. The curve is symmetrical with respect to the ''y''-axis. The curvature radius is <math>r=a\,\operatorname{cot}(x/y)</math>
| |
| | |
| A great implication that the tractrix had was the study of the revolution surface of it around its asymptote: the [[pseudosphere]]. Studied by [[Eugenio Beltrami|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.
| |
| | |
| ==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]] -->
| |
| * 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 <math>a</math>.
| |
| * The [[arc length]] of one branch between ''x'' = ''x''<sub>1</sub> and ''x'' = ''x''<sub>2</sub> is <math>a \ln \frac{x_1}{x_2}</math>
| |
| * The area between the tractrix and its asymptote is <math>\pi a^2/2</math> 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>x = a\cosh\frac{y}{a}</math>.
| |
| * The surface of revolution created by revolving a tractrix about its asymptote is a [[pseudosphere]].
| |
| | |
| ==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.<ref>[http://www.volvotreter.de/downloads/Dinsdale_Horns_1.pdf Horn loudspeaker design pp. 4-5. (Reprinted from Wireless World, March 1974)]</ref><br />
| |
| | |
| ==Drawing machines==
| |
| * In October–November 1692, Huygens described three tractrice drawing machines.
| |
| * In 1693 [[Gottfried Wilhelm Leibniz|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 function|hyperbolic]] quadrature.
| |
| * In 1729 [[Johann Poleni]] built a tractional device that enabled [[logarithm]]ic [[Function (mathematics)|function]]s to be drawn.
| |
| | |
| ==See also==
| |
| *[[Dini's surface]]
| |
| *[[Hyperbolic functions]] for tanh, sech, csch, arccosh
| |
| *[[Natural logarithm]] for ln
| |
| *[[Sign function]] for sgn
| |
| *[[Trigonometric function]] for sin, cos, tan, arccot, csc
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * Edward Kasner & James Newman (1940) [[Mathematics and the Imagination]], pp 141–3, [[Simon & Schuster]].
| |
| * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=5, 199 }}
| |
| | |
| ==External links==
| |
| {{commons|Tractrix}}
| |
| * {{MacTutor|class=Curves|id=Tractrix|title=Tractrix}}
| |
| * {{planetmath reference|id=7109|title=Tractrix}}
| |
| * {{planetmath reference|id=7073|title=Famous curves on the plane.}}
| |
| *[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:Curves]]
| |
| [[Category:Mathematical physics]]
| |
Irwin Butts is what my spouse loves to call me although I don't really like being called like that. North Dakota is her beginning place but she will have to move one at home std testing working day or another. Hiring has been my occupation for some time but I've currently applied for an additional one. home std test kit To play baseball is the hobby he will by no means quit doing.
Review my homepage - www.streaming.iwarrior.net