100:
1087:-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
518:
1077:
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
345:
1425:
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
323:
238:
570:
1080:
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.
1111:. 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.
883:
693:
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).
1405:
513:{\displaystyle 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),}
153:
described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, if the coordinates of the object are
23:
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.
1610:
1361:
1277:
Bertotti, Bruno; Catenacci, Roberto; Dappiaggi, Claudio (2007). "Pseudospheres in geometry and physics: from
Beltrami to de Sitter and beyond".
1296:
1245:
1473:... mechanical devices studied ... to solve particular differential equations ... We must recollect Leibniz's 'universal tractional machine'
253:
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The sign before the solution depends whether the puller moves upward or downward. Both branches belong to the tractrix, meeting at the
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1335:
20:
792:
1548:
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1281:. Ist. Lombardo Accad. Sci. Lett. Incontr. Studio. Vol. 39. LED–Ed. Univ. Lett. Econ. Diritto, Milan. pp. 165–194.
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A great mathematician of the nineteenth century. Papers in honor of
Eugenio Beltrami (1835–1900) (Italian)
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is the length of the pulling thread (4 in the example at right). Then the puller starts to move along the
1674:
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245:
1443:"Recognition and Wonder – Huygens, Tractional Motion and Some Thoughts on the History of Mathematics"
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Due to the geometrical way it was defined, the tractrix has the property that the segment of its
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Perks, John (1706). "The construction and properties of a new quadratrix to the hyperbola".
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The function admits a horizontal asymptote. The curve is symmetrical with respect to the
244:. Writing that the slope of thread equals that of the tangent to the curve leads to the
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In
October–November 1692, Christiaan Huygens described three tractrix-drawing machines.
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The essential property of the tractrix is constancy of the distance between a point
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The surface of revolution created by revolving a tractrix about its asymptote is a
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Beltrami, E. (1868). "Saggio di interpretazione della geometria non euclidea".
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devised a "universal tractional machine" which, in theory, could integrate any
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depends on the direction (positive or negative) of the movement of the puller.
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1240:(revised, 3rd ed.). Springer Science & Business Media. p. 345.
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Horn loudspeaker design pp. 4–5. (Reprinted from
Wireless World, March 1974)
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From Logic to
Practice: Italian Studies in the Philosophy of Mathematics
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or circular tooth shapes, which cause significant sliding and friction.
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892:, between the asymptote and the point of tangency, has constant length
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318:{\displaystyle {\frac {dy}{dx}}=\pm {\frac {\sqrt {a^{2}-x^{2}}}{x}}}
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axis in the positive direction. At every moment, the thread will be
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652:, homogeneous string attached to two points that is subjected to a
16:
Curve traced by a point on a rod as one end is dragged along a line
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790:
773:. The idea was carried further by Kasner and Newman in their book
48:
1330:. Dover Books on Mathematics. Courier Corporation. p. 141.
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233:{\displaystyle y+\operatorname {sign} (y){\sqrt {a^{2}-x^{2}}},}
753:
A great implication that the tractrix had was the study of its
1137:
A history of all these machines can be seen in an article by
71:
to the initial line between the object and the puller at an
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The tractrix might be regarded in a multitude of ways:
565:{\displaystyle a\operatorname {arsech} {\frac {x}{a}},}
122:
in the example shown at right), and the puller at the
1073:
In 1927, P. G. A. H. Voigt patented a
813:
539:
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The first term of this solution can also be written
348:
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along which an object moves, under the influence of
1118:built a tractional machine in order to realise the
706:rolling on a straight line, with its center at the
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317:
232:
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1064:; it cannot be defined by a polynomial equation.
807:The curve can be parameterised by the equation
1467:Milici, Pietro (2014). Lolli, Gabriele (ed.).
712:axis, intersects perpendicularly at all times.
637:rolling (without skidding) on a straight line.
8:
648:function, which describes a fully flexible,
878:{\displaystyle x=t-\tanh(t),y=1/{\cosh(t)}}
765:in 1868, as a surface of constant negative
1394:. McGraw Hill Book Company. p. 20.43.
968:between the tractrix and its asymptote is
1406:"Gates Powergrip GT3 Drive Design Manual"
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779:, where they show a toy train dragging a
610:on the curve and the intersection of the
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1611:MacTutor History of Mathematics Archive
1362:MacTutor History of Mathematics Archive
1226:
1129:built a tractional device that enabled
769:, the pseudosphere is a local model of
1547:Kasner, Edward; Newman, James (1940).
1320:Kasner, Edward; Newman, James (2013).
1530:Epistolarum mathematicanim fasciculus
1436:
1434:
656:field. The catenary has the equation
7:
696:It is a (non-linear) curve which a
1659:Module: Leibniz's Pocket Watch ODE
103:Tractrix with object initially at
14:
1570:A Catalog of Special Plane Curves
1109:first order differential equation
63:attached to a pulling point (the
1441:Bos, H. J. M. (1989).
110:Suppose the object is placed at
1574:. Dover Publications. pp.
1550:Mathematics and the Imagination
1327:Mathematics and the Imagination
776:Mathematics and the Imagination
720:-axis. The curvature radius is
43: 'to pull, drag'; plural:
1008:of the tractrix (that is, the
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832:
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83:in 1670, and later studied by
1:
79:. It was first introduced by
1566:Lawrence, J. Dennis (1972).
1237:Mathematics and Its History
989:, which can be found using
328:with the initial condition
1696:
1487:Philosophical Transactions
783:to generate the tractrix.
1392:Handbook of Metal Forming
1105:Gottfried Wilhelm Leibniz
757:about its asymptote: the
581:inverse hyperbolic secant
75:speed. It is therefore a
1616:University of St Andrews
1367:University of St Andrews
1234:Stillwell, John (2010).
1012:of the tractrix) is the
1532:. p. letter no. 7.
1267:Giornale di Matematiche
1195:Trigonometric functions
95:Mathematical derivation
1499:10.1098/rstl.1706.0017
903:of one branch between
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803:
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234:
107:
24:
1528:Poleni, John (1729).
1131:logarithmic functions
1069:Practical application
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755:surface of revolution
602:Basis of the tractrix
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515:
320:
246:differential equation
235:
102:
22:
1680:Mathematical physics
1602:Robertson, Edmund F.
1555:Simon & Schuster
1390:Lange, Kurt (1985).
1353:Robertson, Edmund F.
1156:Hyperbolic functions
1062:transcendental curve
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537:
346:
254:
176:
1600:O'Connor, John J.;
1415:. 2014. p. 177
1351:O'Connor, John J.;
1254:extract of page 345
771:hyperbolic geometry
633:of the center of a
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242:Pythagorean theorem
55:, when pulled on a
1060:The tractrix is a
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767:Gaussian curvature
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510:
355:
339:. Its solution is
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89:Christiaan Huygens
67:) that moves at a
25:
1413:Gates Corporation
1298:978-88-7916-359-0
1247:978-1-4419-6052-8
1177:Natural logarithm
995:Mamikon's theorem
635:hyperbolic spiral
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1471:. Springer.
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1417:. Retrieved
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1278:
1275:As cited by
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1139:H. J. M. Bos
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1133:to be drawn.
1085:toothed belt
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1419:17 November
1122:quadrature.
1089:trapezoidal
1029:cosh
1020:) given by
1018:chain curve
991:integration
669:cosh
170:-coordinate
87:(1676) and
69:right angle
1669:Categories
1642:PlanetMath
1628:PlanetMath
1623:"Tractrix"
1606:"Tractrix"
1557:. p.
1541:References
1357:"Tractrix"
1120:hyperbolic
1116:John Perks
901:arc length
787:Properties
729:cot
700:of radius
640:It is the
629:It is the
583:function.
45:tractrices
35:(from
1661:at PHASER
1654:MathWorld
1515:186211499
1426:catenary.
1078:tractrix.
863:
830:
824:−
650:inelastic
620:asymptote
618:with the
547:
488:−
473:−
452:−
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413:±
384:−
357:∫
297:−
281:±
213:−
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1650:Tractrix
1456:: 65–76.
1450:Euclides
1145:See also
1125:In 1729
1114:In 1706
1103:In 1693
1014:catenary
1002:envelope
974:π
796:Catenary
646:catenary
642:involute
91:(1693).
53:friction
33:tractrix
29:geometry
1559:141–143
1307:2374676
1047:
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1010:evolute
1006:normals
1004:of the
986:
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890:tangent
800:evolute
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240:by the
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1244:
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1172:arcosh
698:circle
590:point
577:arsech
575:where
544:arsech
165:, the
124:origin
120:(4, 0)
105:(4, 0)
1511:S2CID
1503:JSTOR
1446:(PDF)
1409:(PDF)
1283:arXiv
1221:Notes
631:locus
337:) = 0
126:, so
59:by a
49:curve
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37:Latin
1580:ISBN
1421:2017
1332:ISBN
1293:ISBN
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1197:for
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1168:csch
1164:sech
1160:tanh
1158:for
1016:(or
1000:The
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588:cusp
186:sign
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1199:sin
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993:or
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