查看“︁波塞利耶-利普金机械”︁的源代码

[[File:Peaucellier linkage animation.gif|缩略图|波塞利耶-利普金机械（相同颜色的线具有相同的长度）]]
'''波塞利耶-利普金机械'''，发明于1864年，屬於平面[[连杆机构]]，是第一個真正可以將轉動運動轉換為直線運動的平面[[直線運動機構]]，它以法国陆军军官Charles-Nicolas Peaucellier（1832-1913）和立陶宛犹太人Yom Tov Lipman Lipkin（1846-1876，著名拉比Israel Salanter的兒子）的名字命名<ref>{{cite web |url=http://kmoddl.library.cornell.edu/tutorials/11/ |title=Mathematical tutorial of the Peaucellier–Lipkin linkage |publisher=Kmoddl.library.cornell.edu |accessdate=2011-12-06 |archive-date=2014-09-06 |archive-url=https://web.archive.org/web/20140906051012/http://kmoddl.library.cornell.edu/tutorials/11/ |dead-url=no }}</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 |accessdate=2011-12-06 |archive-date=2011-12-01 |archive-url=https://web.archive.org/web/20111201225836/http://kmoddl.library.cornell.edu/tutorials/04/ |dead-url=no }}</ref>。

在此機構發明之前，在沒有參考導軌的情形下，沒有平面機構可以將直線運動完美的轉換為轉動運動。1864年時，所有的動力來源是來自[[蒸汽机]]，其中有[[活塞]]，由汽缸施力，往上或往下運動。活塞和汽缸需要有良好的密封特性，讓蒸汽机中的蒸汽可以維持在汽缸內，不會因為漏氣而降低能量輸出的效率。活塞和汽缸維持密封的作法是讓活塞維持和汽缸壁平行的直線運動。因此如何讓活塞的直線運動轉換為旋轉運動就變的非常重要，大部份的蒸氣機應用都是旋轉運動。

波塞利耶-利普金机械的數學和圓的{{le|反演幾何|inversive geometry}}有關。

==薩魯斯連桿機構==
在波塞利耶-利普金机械之前，有另外一個立體的直線運動機構，稱為{{le|薩魯斯連桿機構|Sarrus linkage}}，比波塞利耶-利普金机械早11年發明，是由一組以樞紐相連的長方形組成。長方形之間可以以樞紐為軸旋轉，而長方形上的頂點會直線運動。薩魯斯連桿機構屬於立體的空間機構。

==幾何==
[[File:PeaucellierApparatus.PNG|thumb|right|波塞利耶-利普金机械的幾何圖]]
在波塞利耶-利普金机械的幾何圖中，有六個固定長度的桿：OA, OC, AB, BC, CD, DA。OA和OC長度相同，而AB、BC、CD和DA的長度也都相同，形成[[菱形]]。O點是固定點。若B點限制在一個圓的圓周上運動（例如以OB為直徑，通過O和B二點的圓，圖中紅色的圖）。D點會延著直線運動（圖中的藍線）。若點B限制在一直線上運動（不通過O點的直線），則D點會在圓周上運動（通過O點的圓周）。

==數學證明==

===共線===
首先，需要證明O點、B點和D點[[共線 (幾何)|共線]]。這可以用觀察的方式得知，連桿是兩側對稱的，以直線OD為對稱軸，因此B一定在此線上。

若要用正式的方式證明。因為邊BD和自身相等，邊BA和邊BC相等，邊AD和邊CD相等，因此三角形BAD和三角形BCD全等，角BAD和角BCD相等。

接下來要證明三角形OBA和三角形OBC全等。因為線OA和線OC相等，邊OB和自身相等，邊BA和邊BC相等，因此二三角形全等。角OBA和角OBC相等。

以下四個角的和是一個圓角，因此
:∠OBA + ∠ABD + ∠DBC + ∠CBO = 360°

但因為三角形的全等，角OBA = 角OBC，角DBA = 角DBC，因此
:2 × ∠OBA + 2 × ∠DBA = 360°
:∠OBA + ∠DBA = 180°

因此，點O、B、D共線。

===反演點===
令點P為線段AC和線段BD的交點。因為ABCD是菱形，P會是線段AC和線段BD的[[中點]]，因此，線段BP和線段PD等長。

因為邊BP和邊DP相等，邊AP和自身相等，邊AB和邊AD相同，因此三角形BPA和三角形DPA全等。因此角BPA等於角DPA。但因為角BPA + 角DPA = 180°，因此角BPA和角DPA都是90°。

令：
:<math> x = \ell_{BP} = \ell_{PD} </math>
:<math> y = \ell_{OB} </math>
:<math> h = \ell_{AP} </math>

則：
:<math>\ell_{OB}\cdot \ell_{OD}=y(y+2x)=y^2+2xy </math>
:<math>{\ell_{OA}}^2 = (y + x)^2 + h^2</math> <small>（因為[[畢氏定理]]）</small>
:<math>{\ell_{OA}}^2 = y^2 + 2xy + x^2 + h^2</math>
:<math>{\ell_{AD}}^2 = x^2 + h^2</math> <small>（因為畢氏定理）</small>
:<math>{\ell_{OA}}^2 - {\ell_{AD}}^2 = y^2 + 2xy = \ell_{OB} \cdot \ell_{OD}</math>

因為OA和AD的長度固定,，因此OB和OD的乘積為定值：
:<math>\ell_{OB}\cdot \ell_{OD} = k^2 </math>

又因為O點、B點和D點共線，因此D點是B點相對圓(O,''k'')（圓心在O點，半徑為k）的反演點。

===反演幾何===
透過{{le|反演幾何|inversive geometry}}的特性，因為點D的軌跡是點B軌跡的反演。若B的軌跡是通過反演中心O的圓，則點D的軌跡會是一直線。若點B的軌跡是不通過點O的直線，則點D的軌跡是通過點O的圓。[[Q.E.D.]]

===典型的主動件===
[[File:The Peaucellier-Lipkin linkage with a rocker-slider four-bar as its driver.gif|thumb|right|圖中的滑塊搖桿四連桿是波塞利耶-利普金机械的輸入]]
波塞利耶-利普金机械有許多的反演機構。其中一個如圖所示，以滑塊搖桿四連桿（ rocker-slider four-bar）為輸入，若要再細分，滑塊為輸入，使得搖桿以及波塞利耶-利普金机械轉動。

==展覽物==
在荷蘭埃因霍溫的永久展覽品中，有展覽物就是以此機構為主題。此展覽物大小為{{convert|22|x|15|x|16|m}}，重{{convert|6600|kg}}，遊客可以透過控制盤操作<ref name="Schoofs">{{cite web|title=Just because you are a character, doesn't mean you have character|url=https://ivoschoofs.com/project/just-because-you-are-a-character-doesnt-mean-you-have-character/|website=Ivo Schoofs|accessdate=2017-08-14|archive-date=2020-12-02|archive-url=https://web.archive.org/web/20201202060715/https://ivoschoofs.com/project/just-because-you-are-a-character-doesnt-mean-you-have-character/|dead-url=no}}</ref>。


==相關條目==
*[[哈特倒置器]]
*[[连杆机构]]

==參考資料==
{{reflist}}

==文獻==
* {{citation | last = Ogilvy | first = C. S. | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7 | pages = [https://archive.org/details/excursionsingeom0000ogil/page/46 46&ndash;48] | ref = none | url = https://archive.org/details/excursionsingeom0000ogil/page/46 }}
* {{cite book|last=Bryant|first=John|title=How round is your circle? : where engineering and mathematics meet|url=https://archive.org/details/howroundisyourci00brya|year=2008|publisher=Princeton University Press|location=Princeton|isbn=978-0-691-13118-4|author2=Sangwin, Chris|pages=[https://archive.org/details/howroundisyourci00brya/page/n55 33]–38; 60–63}} — proof and discussion of Peaucellier–Lipkin linkage, mathematical and real-world mechanical models
* {{cite book | title = Geometry Revisited | url = https://archive.org/details/geometryrevisite00coxe | url-access = limited | author = Coxeter HSM, Greitzer SL| year = 1967 | publisher = Mathematical Association of America | location = Washington | isbn = 978-0-88385-619-2 | pages = [https://archive.org/details/geometryrevisite00coxe/page/n119 108]&ndash;111}} (and references cited therein)
* Hartenberg, R.S. & J. Denavit (1964) [http://kmoddl.library.cornell.edu/bib.php?m=23 Kinematic synthesis of linkages] {{Wayback|url=http://kmoddl.library.cornell.edu/bib.php?m=23 |date=20110519063139 }}, pp 181&ndash;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&ndash;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 = [https://archive.org/details/penguindictionar0000well/page/120 120] | url = https://archive.org/details/penguindictionar0000well/page/120 }}

==外部連結==
{{commons category|Peaucellier–Lipkin linkage}}
* [https://web.archive.org/web/20080125112743/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] {{Wayback|url=http://kmoddl.library.cornell.edu/tutorials/04/ |date=20111201225836 }}
* [http://xahlee.org/SpecialPlaneCurves_dir/ggb/Peaucellier_Linkage_line.html Interactive Java Applet with proof.] {{Wayback|url=http://xahlee.org/SpecialPlaneCurves_dir/ggb/Peaucellier_Linkage_line.html |date=20120627023904 }}
* [http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/index.html Java animated Peaucellier&ndash;Lipkin linkage] {{Wayback|url=http://www.math.toronto.edu/~drorbn/People/Eldar/thesis/index.html |date=20100923234636 }}
* [http://bible.tmtm.com/wiki/LIPKIN_%28Jewish_Encyclopedia%29 Jewish Encyclopedia article on Lippman Lipkin] {{Wayback|url=http://bible.tmtm.com/wiki/LIPKIN_%28Jewish_Encyclopedia%29 |date=20070305232921 }} and his father Israel Salanter
*[https://web.archive.org/web/20061208133000/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] {{Wayback|url=http://mw.concord.org/modeler1.3/mirror/mechanics/peaucellier.html |date=20110725190957 }} using the Molecular Workbench software
*[http://mathworld.wolfram.com/HartsInversor.html A related linkage] {{Wayback|url=http://mathworld.wolfram.com/HartsInversor.html |date=20200115185706 }} 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)] {{Wayback|url=http://vamfun.wordpress.com/2011/07/13/team-1508a-vex-peaucellier-lift-roundup-youtube-video/ |date=20190705053538 }}
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