201.80.232.111: /* The mathematical form of the Rasch model for dichotomous data */

2014-12-13T20:45:53Z

The mathematical form of the Rasch model for dichotomous data

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	~~I am 39~~ years ~~old~~ and ~~my name~~ is ~~Wallace Mccallister~~. ~~I life~~ in ~~Humtschach~~ (~~Austria~~).<br><br>~~Look into~~ my ~~homepage~~; [http://~~Www~~.~~arsoperandi~~.com/~~2009~~/04/~~nueva~~-~~cabecera~~-~~gomez~~-~~losada.html How To Get Free Fifa 15 Coins~~]		[[File:Peaucellier linkage animation.gif\|frame\|right\|Peaucellier-Lipkin linkage:<br>bars of identical colour are of equal length]]
			The '''Peaucellier–Lipkin linkage''' (or '''Peaucellier–Lipkin cell''', or '''Peaucellier–Lipkin Inversor'''), invented in 1864, was the first planar [[straight line mechanism]] -- the first planar [[linkage (mechanical)\|linkage]] capable of transforming [[rotary motion]] into perfect [[straight-line motion]], and vice versa. It is named after [[Charles-Nicolas Peaucellier]] (1832–1913), a French army officer, and [[Yom Tov Lipman Lipkin]], a [[Lithuanian Jew]] and son of the famed Rabbi [[Israel Salanter]].<ref>{{cite web\|url=http://kmoddl.library.cornell.edu/tutorials/11/ \|title=Mathematical tutorial of the Peaucellier–Lipkin linkage \|publisher=Kmoddl.library.cornell.edu \|date= \|accessdate=2011-12-06}}</ref><ref>{{cite web\|last=Taimina \|first=Daina \|url=http://kmoddl.library.cornell.edu/tutorials/04/ \|title=How to draw a straight line by Daina Taimina \|publisher=Kmoddl.library.cornell.edu \|date= \|accessdate=2011-12-06}}</ref>

			Until this invention, no planar method existed of producing straight motion without reference guideways, making the linkage especially important as a machine component and for manufacturing. In particular, a [[piston]] head needs to keep a good seal with the shaft in order to retain the driving (or driven) medium. The Peaucellier linkage was important in the development of the [[steam engine]].

			The mathematics of the Peaucellier–Lipkin linkage is directly related to the [[inversive geometry\|inversion]] of a circle.

			==Earlier Sarrus linkage==
			There is an earlier straight-line mechanism, whose history is not well known, called [[Sarrus linkage]]. This linkage predates the Peaucellier–Lipkin linkage by 11 years and consists of a series of hinged rectangular plates, two of which remain parallel but can be moved normally to each other. Sarrus' linkage is of a three-dimensional class sometimes known as a [[space crank]], unlike the Peaucellier–Lipkin linkage which is a planar mechanism.

			==Geometry==
			[[File:PeaucellierApparatus.PNG\|thumb\|right\|Geometric diagram of a Peaucellier linkage]]
			In the geometric diagram of the apparatus, six bars of fixed length can be seen: OA, OC, AB, BC, CD, DA. The length of OA is equal to the length of OC, and the lengths of AB, BC, CD, and DA are all equal forming a [[rhombus]]. Also, point O is fixed. Then, if point B is constrained to move along a circle (shown in red) which passes through O, then point D will necessarily have to move along a straight line (shown in blue). On the other hand, if point B were constrained to move along a line (not passing through O), then point D would necessarily have to move along a circle (passing through O).

			==Mathematical proof of concept==

			===Collinearity===
			First, it must be proven that points O, B, D are [[Line (geometry)\|collinear]].

			Triangles BAD and BCD are congruent because side BD is congruent to itself, side BA is congruent to side BC, and side AD is congruent to side CD. Therefore angles ABD and CBD are equal.

			Next, triangles OBA and OBC are congruent, since sides OA and OC are congruent, side OB is congruent to itself, and sides BA and BC are congruent. Therefore angles OBA and OBC are equal.

			:angle OBA + angle ABD + angle DBC + angle CBO = 360°

			but angle OBA = angle OBC and angle DBA = angle DBC, thus
			:2 × angle OBA + 2 × angle DBA = 360°
			:angle OBA + angle DBA = 180°

			therefore points O, B, and D are collinear.

			===Inverse points===
			Let point P be the intersection of lines AC and BD. Then, since ABCD is a [[rhombus]], P is the [[midpoint]] of both line segments BD and AC. Therefore length BP = length PD.

			Triangle BPA is congruent to triangle DPA, because side BP is congruent to side DP, side AP is congruent to itself, and side AB is congruent to side AD. Therefore angle BPA = angle DPA. But since angle BPA + angle DPA = 180°, then 2 × angle BPA = 180°, angle BPA = 90°, and angle DPA = 90°.

			Let:
			:<math>\begin{align}
			x &= \ell_{BP} = \ell_{PD} \\
			y &= \ell_{OB} \\
			h &= \ell_{AP}
			\end{align}</math>

			Then:
			:<math>\ell_{OB}\cdot \ell_{OD}=y(y+2x)=y^2+2xy </math>
			:<math>{\ell_{OA}}^2 = (y + x)^2 + h^2</math> <small>(due to the [[Pythagorean theorem]])</small>
			:<math>{\ell_{AD}}^2 = x^2 + h^2</math> <small>(Pythagorean theorem)</small>
			:<math>{\ell_{OA}}^2 - {\ell_{AD}}^2 = y^2 + 2xy = \ell_{OB} \cdot \ell_{OD}</math>

			Since OA and AD are both fixed lengths, then the product of OB and OD is a constant:
			:<math>\ell_{OB}\cdot \ell_{OD} = k^2 </math>

			and since points O, B, D are collinear, then D is the inverse of B with respect to the circle (O,''k'') with center O and radius ''k''.

			===Inversive geometry===
			Thus, by the properties of [[inversive geometry]], since the figure traced by point D is the inverse of the figure traced by point B, if B traces a circle passing through the center of inversion O, then D is constrained to trace a straight line. But if B traces a straight line not passing through O, then D must trace an arc of a circle passing through O. ''[[Q.E.D.]]''

			===A typical driver===
			[[File:The Peaucellier-Lipkin linkage with a rocker-slider four-bar as its driver.gif\|thumb\|right\|Slider-Rocker Four-Bar <br>acts as the driver of the Peaucellier-Lipkin linkage]]
			Peaucellier–Lipkin linkages (PLLs) may have several inversions. A typical example is shown in the opposite figure, in which a rocker-slider four-bar serves as the input driver. To be precise, the slider acts as the input, which in turn drives the right grounded link of the PLL, thus driving the entire PLL.

			===Historical notes===
			[[James Joseph Sylvester\|Sylvester]] (Collected Works, Vol. 3 Paper 2 ) writes that when he showed a model to [[Lord Kelvin\|Kelvin]], he 'nursed it as if it had been his own child, and when a motion was made to relieve him of it, replied "No! I have not had nearly enough of it—it is the most beautiful thing I have ever seen in my life"'.
			<!-- Cette question a été communiquée, au nom du commandant Peaucellier, par
			M. Mannheim, à la séance de la Société Philomathique de Paris du 20 juillet 1867.
			M. Peaucellier l’avait déjà posée dans les Nouvelles Annales de Mathématique, 2e série, t. III, p. 414, 1864; il en a, de plus, appliqué le principe à un appareil pour mesurer les distances, qui se trouve décrit dans le Mémorial de l’Officier du Génie, no 18, année 1868. Ces détails historiques sont nécessaires, parce que M. Lipkin donne, en août 1871, le même théorème dans la Revue universelle des Mines et de la Métallurgie de Liège, 15e année, t. XXX, 4e livraison, p. 149 et 150.

			See: http://hal.archives-ouvertes.fr/docs/00/23/68/20/PDF/ajp-jphystap_1873_2_130_1.pdf -->

			==See also==
			*[[Hart's inversor]]

			==References==
			{{reflist}}

			==Bibliography==
			* {{citation \| author = Ogilvy CS \| year = 1990 \| title = Excursions in Geometry \| publisher = Dover \| isbn = 0-486-26530-7 \| pages = 46–48}}
			* {{cite book\|last=Bryant\|first=John\|title=How round is your circle? : where engineering and mathematics meet\|year=2008\|publisher=Princeton University Press\|location=Princeton\|isbn=978-0-691-13118-4\|coauthors=Sangwin, Chris\|pages=33–38; 60–63}} — proof and discussion of Peaucellier–Lipkin linkage, mathematical and real-world mechanical models
			* {{cite book \| title = Geometry Revisited \| author = [[Harold Scott MacDonald Coxeter\|Coxeter HSM]], [[S. L. Greitzer\|Greitzer SL]]\| year = 1967 \| publisher = [[Mathematical Association of America\|MAA]] \| location = [[Washington, D.C.\|Washington]] \| isbn = 978-0-88385-619-2 \| pages = 108–111}} (and references cited therein)
			* Hartenberg, R.S. & J. Denavit (1964) [http://kmoddl.library.cornell.edu/bib.php?m=23 Kinematic synthesis of linkages], pp 181–5, New York: McGraw-Hill, weblink from [[Cornell University]].
			* {{cite book \| author = Johnson RA \| year = 1960 \| title = Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle \| edition = reprint of 1929 edition by Houghton Miflin \| publisher = Dover Publications \| location = New York \| isbn = 978-0-486-46237-0 \| pages = 46–51}}
			* {{cite book \| author = Wells D \| year = 1991 \| title = The Penguin Dictionary of Curious and Interesting Geometry \| publisher = Penguin Books \| location = New York \| isbn = 0-14-011813-6 \| page = 120}}

			==External links==
			* [http://www.howround.com/ How to Draw a Straight Line, online video clips of linkages with interactive applets.]
			* [http://kmoddl.library.cornell.edu/tutorials/04/ How to Draw a Straight Line, historical discussion of linkage design]
			* [http://xahlee.org/SpecialPlaneCurves_dir/ggb/Peaucellier_Linkage_line.html Interactive Java Applet with proof.]
			* [http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/index.html Java animated Peaucellier–Lipkin linkage]
			* [http://bible.tmtm.com/wiki/LIPKIN_%28Jewish_Encyclopedia%29 Jewish Encyclopedia article on Lippman Lipkin] and his father [[Yisrael Lipkin Salanter\|Israel Salanter]]
			*[http://www.ies.co.jp/math/java/geo/hantenki/hantenki.html Peaucellier Apparatus] features an interactive applet
			*[http://mw.concord.org/modeler1.3/mirror/mechanics/peaucellier.html A simulation] using the Molecular Workbench software
			*[http://mathworld.wolfram.com/HartsInversor.html A related linkage] called Hart's Inversor.
			*[http://vamfun.wordpress.com/2011/07/13/team-1508a-vex-peaucellier-lift-roundup-youtube-video/ Modified Peaucellier robotic arm linkage (Vex Team 1508 video)]
			{{Piston engine configurations\|state=uncollapsed}}

			{{DEFAULTSORT:Peaucellier-Lipkin Linkage}}
			[[Category:Linkages]]
			[[Category:Articles containing proofs]]
			[[Category:Linear motion]]

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Rasch model - Revision history

201.80.232.111: /* The mathematical form of the Rasch model for dichotomous data */

en>WeijiBaikeBianji: Reverted good faith edits by 49.128.169.68 (talk): Signatures belong on talk pages, not in article text. (TW)

en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Comparison