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opposite to the sense of rotation of the trammel. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. This is compounded by high forces owing to the short throw of the crank of only 1/4 the travel of the pivots.
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The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. This circle corresponds
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The point midway between the pivots orbits in a circle around the point where the channels cross. This circle is also a special case of an ellipse. Here the axes are of equal length. The diameter of the circle is equal to the distance between the pivots. The direction of travel around the orbit is
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54:. It consists of two shuttles which are confined to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at adjustable positions along the rod.
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As the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. The motion of the rod is termed elliptical motion. The semi-axes
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with two sliders and two pivots, and is special case of the more general oblique trammel. The axes constraining the pivots do not have to be perpendicular and the points
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Loci of some points along and beyond a trammel of
Archimedes, the green circle being the loci of its midpoint –
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of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots.
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An ellipsograph is a trammel of
Archimedes intended to draw, cut, or machine ellipses, e.g. in
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or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife,
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Practical Conic
Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas
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The history of such ellipsographs is not certain, but they are believed to date back to
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handle, and the positions of the sliding shuttles along the rod are usually fixed.
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are adjustable, so that the size and shape of the ellipse can be varied.
465:{\displaystyle {\frac {x^{2}}{(AB+BC)^{2}}}+{\frac {y^{2}}{(BC)^{2}}}=1}
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US-Patent 4306598 for ellipse cutting guide allowing small ellipses
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Wetzel, John E. (February 2010). "An
Ancient Elliptic Locus".
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for an ellipse in canonical position. The further equation
116:). In these toys the drafting instrument is replaced by a
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move the pointer over the diagram to move the trammel
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Musée d'histoire des sciences de la Ville de Genève
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50:. One common form of ellipsograph is known as the
248:axes, respectively. When the rod makes an angle
752:Mechanisms for the Generation of Plane Curves
478:The trammel of Archimedes is an example of a
8:
229:, respectively. Let us assume that sliders
773:Video of various trammel designs in action
544:, etc.) attached to the rod. Usually the
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807:An exploration of a generalized trammel.
185:Trammel of Archimedes with three sliders
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310:{\displaystyle x=(AB+BC)\cos \theta \,}
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360:These are in the form of the standard
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145:Trammel of Archimedes as ellipsograph
34:Trammel of Archimedes animated model.
7:
640:Mathematical Association of America
494:can form a triangle. The resulting
205:be the pivots of the sliders. Let
350:{\displaystyle y=BC\sin \theta \,}
25:
197:be the outer end of the rod, and
84:"Bullshit Grinder" toy (ca. 1960)
562:and perhaps even to the time of
256:-axis, the coordinates of point
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46:that generates the shape of an
817:Secrets of the Nothing Grinder
783:Photo of a Kentucky Do-Nothing
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798:"Wonky Trammel of Archimedes"
680:American Mathematical Monthly
630:Schwartzman, Steven (1996).
529:Ellipsograph on display at
69:to the smaller circle in a
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790:of a Do-Nothing made from
88:Versions are also made as
27:Ellipse-drawing mechanism
842:Mechanisms (engineering)
778:Cutting ellipses in wood
693:10.4169/000298910x476068
634:The Words of Mathematics
96:(sold under the name of
754:. Pergamon Press 1964,
740:restricted online copy
729:. Courier Dover 2003,
657:restricted online copy
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475:is immediate as well.
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213:be the distances from
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847:Linkages (mechanical)
611:Scott Russell linkage
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502:is still an ellipse.
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52:trammel of Archimedes
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18:Trammel of Archimedes
750:I. I. Artobolevskii
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362:parametric equations
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98:Kentucky do-nothings
106:do nothing machines
803:2012-02-20 at the
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518:(ca. 1900) now at
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307:
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36:
820:YouTube video by
735:978-0-486-42876-5
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114:bullshit grinders
16:(Redirected from
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862:Educational toys
837:Traditional toys
737:, pp. 4–5 (
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767:External links
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760:978-1483120003
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687:(2): 161–167.
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662:Google Books
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596:Hypotrochoid
576:Beam compass
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516:ellipsograph
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40:ellipsograph
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743:, p. 4, at
601:Tusi couple
591:Hypocycloid
520:Smithsonian
124:Mathematics
71:Tusi couple
831:Categories
822:Mathologer
720:References
564:Archimedes
246:coordinate
709:117701083
546:distances
344:θ
341:
304:θ
301:
252:with the
44:mechanism
801:Archived
570:See also
560:Proclus
514:Wooden
157:Diagram
130:Concept
48:ellipse
794:bricks
758:
733:
707:
699:
646:
638:. The
542:router
788:Video
705:S2CID
697:JSTOR
617:Notes
496:locus
118:crank
112:, or
42:is a
792:Lego
756:ISBN
731:ISBN
644:ISBN
551:and
538:wood
490:and
241:and
233:and
221:and
209:and
193:Let
90:toys
61:and
689:doi
685:117
498:of
338:sin
298:cos
225:to
217:to
92:or
38:An
833::
703:.
701:10
695:.
683:.
669:^
642:.
566:.
486:,
211:BC
207:AB
201:,
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762:.
747:)
711:.
691::
664:)
654:(
652:.
553:b
549:a
500:C
492:C
488:B
484:A
460:1
457:=
449:2
445:)
441:C
438:B
435:(
429:2
425:y
419:+
411:2
407:)
403:C
400:B
397:+
394:B
391:A
388:(
382:2
378:x
335:C
332:B
329:=
326:y
295:)
292:C
289:B
286:+
283:B
280:A
277:(
274:=
271:x
258:C
254:x
250:θ
243:x
239:y
235:B
231:A
227:C
223:B
219:B
215:A
203:B
199:A
195:C
63:b
59:a
20:)
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