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Topic: Tractrix

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	PlanetMath: tractrix (Site not responding. Last check: 2007-10-16)
		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.
		The differential equation of the orthogonal curves of the tractrix is
		This is version 22 of tractrix, born on 2005-05-24, modified 2006-03-27.
	planetmath.org /encyclopedia/Tractrix.html (302 words)

	Tractrix
		The evolute of a tractrix is a catenary.
		Among the properties of the tractrix are the fact that the length of a tangent from its point of contact to an asymptote is constant.
		The area between the tractrix and its asymptote is finite.
	www-groups.dcs.st-and.ac.uk /~history/Curves/Tractrix.html (211 words)

	Tractrix (Site not responding. Last check: 2007-10-16)
		Tractrix - Claude Perrault is not a well known mathematical name, in fact there is no record of his ever proving any mathematical theorem.
		Perrault's question would today be called a differential equation, and the name of the curve is the tractrix, which is drawn from the Latin root tractus, that which is pulled or drawn along.
		A tractrix is also known to be orthogonal (perpendicular) to all the set of circles with a common radius along a straight line.
	www.pballew.net /tractrix.html (466 words)

	tractrix
		Perrault is not a giant in the annals of mathematics; in fact, he trained as a doctor and gained a minor reputation as an architect and an anatomist before dying in unusual style as a result of an infection he caught while dissecting a camel.
		The first known solution was given in a letter to a friend in 1693 by Christiaan Huygens, who also coined the name "tractrix" from the Latin tractus for something that is pulled along.
		The surface of revolution of the tractrix is the pseudosphere, which is the classic model for hyperbolic geometry and one possible three-dimensional analog for the shape of the four-dimensional spacetime in which we live.
	www.daviddarling.info /encyclopedia/T/tractrix.html (299 words)

	Special Plane Curves: Tractrix
		The evolute of tractrix is catenary, conversely, the involute of catenary is tractrix.
		On the right, the normals of the tractrix is draw, and their envelope forms its evolute.
		The tractrix is orthogonal to a set of circles of constant radius whose centers are on the asymptote of the tractrix; conversely, the curve that cuts 90 degrees into a series of identical circles on a line is the tractrix.
	xahlee.org /SpecialPlaneCurves_dir/Tractrix_dir/tractrix.html (283 words)

	Retrofit valve stem seal assembly - Patent 5601276
		A tractrix is a curve the length of whose tangent, from point of tangency of the curve to the x-axis, is constant.
		This comprises a sliding tractrix surface 7 on the valve stem adaptor 3.
		2, a tractrix is a curve the length of whose tangent, from point of tangency of the curve to the x-axis, is constant.
	www.freepatentsonline.com /5601276.html (2788 words)

	Tractrix Java Applet Class
		Tractrix is a Java applet which makes the pretty swirling patterns that you can see on this page (assuming that you have Java running).
		The Tractrix pattern was named in 1692 by the Dutch mathematician Huygens.
		For a tractrix that does not move, make the increment zero; the maximum is then ignored.
	www.phdcc.com /tractrix.html (404 words)

	Tractrix Article - Study Science Articles
		The tractrix is a curve used to describe mathematical, physical and engineering situations and proceses.
		The essential property of the tractrix is thet the lenght of the tangent to it and the Ox axis remains constant at any given point.
		Orthogonal to a set of circles of constant radius whose centers are on the asymptote of the tractrix; conversely.
	www.1888articles.com /tractrix-0v603dt.html (588 words)

	Tractrix - Wikipedia, the free encyclopedia
		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.
		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 envelope of the normals of the tractrix, that is, the evolute of the tractrix is the catenary (or chain curve) given by
	en.wikipedia.org /wiki/Tractrix (316 words)

	Leonardo Computer Systems Ltd - The CAD Specialists
		TracTrix is very forgiving in that the vectors created can be deleted and re-vectorized, edited and all vector errors corrected before saving in the CAD file format of your choice.
		Many companies employ TracTrix to shorten time to manufacture by allowing sketches and other irregular drawings to be rapidly taken into CAD format for use in numerically controlled milling machines, machining centers, laser, water jet and plasma arc cutters.
		An algorithm in TracTrix is designed to seek the pixels in the raster image and 'follow' them according to the parameters set by the operator.
	www.leonardo-cad.co.uk /exchange/scanning.htm (4751 words)

	Tractrix - Discover spin track and wafer coating.
		The Tractrix™ spin tool is a stand alone, small-footprint track system designed for the spin deposition of photoresist, developer, polymer and other materials common to integrated circuit photolithography.
		Built with the philosophy of "one controller for one track," the Tractrix™ tool is highly reliable.
		Owing to its unique, efficient design, the Tractrix™ system is easier to operate and requires far less maintenance than any other track system utilizing multi-axis robotic transfer.
	www.site.com /main/tractrix.htm (146 words)

	CAD-Software - AutoCAD & IntelliCAD solutions
		TracTrix is very forgiving in that the vectors created can be deleted and re-vectorized, edited and all vector errors corrected before saving in the CAD file format of your choice.
		Many companies employ TracTrix to shorten time to manufacture by allowing sketches and other irregular drawings to be rapidly taken into CAD format for use in numerically controlled milling machines, machining centers, laser, water jet and plasma arc cutters.
		An algorithm in TracTrix is designed to seek the pixels in the raster image and 'follow' them according to the parameters set by the operator.
	www.cad-software.com.au /trix (4879 words)

	Amazon.com: tractrix (Site not responding. Last check: 2007-10-16)
		14 The tractrix as a curve of pursuit.
		Quadratrix of Hippias, the Epitrochoid, Tractrix, and the Cassinian Ovals.
		C is called a tractrix and is clearly given by...
	www.amazon.com /s?ie=UTF8&keywords=tractrix&tag=icongroupinterna&index=blended&link_code=qs&page=1 (1004 words)

	The Exemplar Project
		Enter the Tractrix It was during this period that we heard a full range speaker system employing tractrix horn flares for the midrange and bass drivers.
		And when different types were mixed, such as a VOT bass with a tractrix treble, or a tractrix bass with an exponential treble, their unique signatures were very apparent, and didn't sound well integrated.
		Tractrix Compression Driver Horn The prototype tractrix compression driver horn was also fashioned from wood, covered a narrow 35 degrees in the horizontal, and had a mouth cutoff of 333 Hz.
	www.omegav.ntnu.no /~dunker/jeffjohn.html (1867 words)

	HornCalc Tractrix Horn Calculation Software
		With the tractrix horn, since the waves are hemispherical going through the horn, they exit with very few reflections.
		The curve whose tangent to the centre is of constant length is called the "Tractrix", and may be drawn by drawing a small portion of one tangent after the other.
		In the case of a square horn, the tangent to the corners is longer than that to the middle of the sides and a compromise is unavoidable.
	www.sonicdesign.se /horncalc.html (2112 words)

	Klipsch RF-5 Loudspeakers reviewed on AudioRevolution.com
		The inch Tractrix horn is set above the two vertically arrayed eight-inch Cerametallic drivers in a rear-ported and internally braced MDF cabinet that measures 41.5 inches high, nine inches wide and 14 inches deep, weighing 60 pounds apiece.
		As the sound exits the larger portion of the horn, the rectangular shape is designed to control dispersion, 90 degrees in the vertical, and 60 degrees in the horizontal, which is said to minimize side wall interaction, one of the most offending situations that a speaker owner may confront.
		The eight-inch Tractrix horn uses a very large video-shielded 9.6-ounce magnet that actually fully encloses the rear portion of the motor, which provides more magnet surface area, as opposed to a hollow core magnet that surrounds the motor (the type found in my old Heresys).
	www.avrev.com /equip/klipschfr5/index.html (1839 words)

	VisualCalc
		Another such example is the tractrix, the trajectory of a toy being pulled by a string of constant length by a child walking along a fixed straight line as shown in Figure 4.
		The tractrix and exponential have been studied for centuries, but apparently no one realized that they are related to one another.
		Both the tractrix and the exponential, which have been studied for centuries, turn out to be special cases of a family of curves characterized by the equation _t + _s = constant.
	www.cco.caltech.edu /~mamikon/VisualCalc.html (5469 words)

	www.myspace.com/seedsofcivilization
		Tractrix sends readers on an engaging joyride through a labyrinth of riddles involving the ancient Maya, a secret U.S. military installation in the Nevada desert, and alien visitors from centuries past.
		Tractrix is a fast-paced yarn, based on intriguing historical perspective, and it's sure to keep you guessing and wondering right up until its surprising conclusion.
		In Tractrix the characters explore the mystery behind the Maya of Mexico's Yucatan Peninsula; In the sequel Tsubute, they uncover the secrets of the Yonaguni Monument off southern Japan; In soon-to-be-released novel three, Triangle, the characters explore a submerged city deep in Caribbean waters off the northwestern tip of Cuba.
	www.myspace.com /seedsofcivilization (1584 words)

	Tractrix -- from Wolfram MathWorld
		When a tractrix is rotated around its asymptote, a pseudosphere results.
		For a tractrix, the length of a tangent from its point of contact to an asymptote is constant.
		Gray, A. "The Tractrix" and "The Evolute of a Tractrix is a Catenary." §3.6 and 5.3 in
	mathworld.wolfram.com /Tractrix.html (337 words)

	Atkinsopht/Rowing/Oarblade Path Geometry
		Another way to visualize the zero-slip locus is to consider an ice boat whose oarblades are figure skate blades constrained by their bite in the ice to move only along their chords.
		The zero-slip advance of the shell (the oarlock) per stroke (Sz) is the sum of four distances, for a blade with zero cant angle, as illustrated above.
		When the cant angle is not zero the tractrix becomes "oblique" or skewed and the calculation of values for S2 and S3 becomes more complex than is shown in Figure 1.
	www.atkinsopht.com /row/bladpath.htm (887 words)

	tractrix (Site not responding. Last check: 2007-10-16)
		For the tractrix the length of a tangent from the tangent point to the asymptote is constant, leading to its alternative name of equitangential curve.
		The curve must not be confused with the pursuit curve, while there is no fixed length between dragging and dragged object.
		Beltrami (1868) used the tractrix to rotate around its asymptote, in his study of non-Euclidean geometry.
	www.2dcurves.com /exponential/exponentialt.html (158 words)

	Klipsch Reference RP-5
		The common denominator is a 1-inch titanium dome compression driver coupled to a Tractrix horn.
		Both research and praxis over the past ten years has shown that the Tractrix horn with its large flare rate is ideal for the reproduction of voice.
		The Klipsch Tractrix has been designed to be less beamy than traditional horns and achieves uniform directivity over a 90 by 60 degree field.
	www.enjoythemusic.com /magazine/equipment/1299/klipschrp5.htm (1512 words)

	TracTrix Desktop 2
		TracTrix Desktop 2 is not restricted to swatches.
		TracTrix Desktop 2 includes an expert system for color separation that regroups color variations from scanners.
		TracTrix Desktop 2 features color separation, and is very accurate for small logos and illustrations.
	www.trixsystems.se /ttd20.htm (230 words)

	Curves in a plane
		This locus is called the tractrix associated to the catenary.
		From this, we see that the tangent line in T to the tractrix is orthogonal to the tangent line in P to the catenary.
		The normal in T to the tractrix is tangent to the catenary.
	www.ping.be /~ping1339/curves.htm (2434 words)

	edgarinterview
		With the Tractrix horn, since the waves are spherical going through the horn, they exit the mouth with very few reflections.
		Bruce: The rectangular Tractrix horn always generates a specific sweet point for imaging, but the true Tractrix is a round horn.
		And I designed a Tractrix full-size horn that was about two feet tall and I think about twenty inches wide, which filled in the difference between 150 Hertz and 500 Hertz, and then I put my regular midrange/tweeter on top of that.
	www.positive-feedback.com /Issue4/edgarinterview.htm (13289 words)

	May 23 - Trix Ships TracTrix v5 for Raster to Vector (Site not responding. Last check: 2007-10-16)
		“We see TracTrix as the CAD user’s conversion companion – in addition to converting scans and photos into CAD you can also use your TracTrix for marking up and converting CAD files into formats that can be shared easily with people who don’t have CAD.” says Trix Systems CEO, Jorgen Frilander.
		When asked what’s changed the most since the first introduction of TracTrix, lead developer, Per Hillerstrom, says “Fifteen years ago raster images came from scanners that were fairly simple devices, mostly producing monochrome output.
		For a fully functional trial version of TracTrix visit http://www.trixsystems.com/tractrix.html or call Trix Systems at (978) 256-4445.
	www.tenlinks.com /news/pr/trix/052305_tractrix5.htm (433 words)

	Raster to vector and vector to raster conversion software - TracTrix (Site not responding. Last check: 2007-10-16)
		To convert drawings, TracTrix automatically traces pixels in the lines of the scan and converts them to vectors.
		TracTrix is the easiest and definitely the most cost effective of the raster to vector conversion software product that I've tested.
		Share a TracTrix license with colleagues on the same network.
	www.trixsystems.com /raster-to-vector-conversion-software.html (352 words)

	PHASER: Module: Leibnitz's Pocket Watch ODE
		Geometrically, this problem can be recast as follows: find a curve with the property that at each point on the curve, the tangent line segment between the point of tangency and the x-axis is of constant length a.
		Such a curve is called the tractrix or equitangential curve.
		In differential geometry, a parameterization of the tractrix, for a = 1, is given by the equations
	www.phaser.com /modules/historic/leibniz (780 words)

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