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The Quadruplanar inversor of Sylvester and Kempe is a generalization of Hart's inversor. Like Hart's inversor, is a mechanism that provides a perfect straight line motion without sliding guides.

The mechanism was described in 1875 by James Joseph Sylvester in the journal Nature.^[1]

Like Hart's inversor, it is based on an antiparallelogram but the rather than placing the fixed, input and output points on the sides (dividing them in fixed proportion so they are all similar), Sylvester recognized that the additional points could be displaced sideways off the sides, as long as they formed similar triangles. Hart's original form is simply the degenerate case of triangles with altitude zero.

Gallery

In these diagrams:

The antiparallelogram is highlighted in full opacity links.
Yellow Triangles and Green Triangles are similar.
- Green Triangles are congruent with each other.
- Yellow Triangles are congruent with each other.
Cyan links and Pink links are congruent.
Dashed links are additional appendages to allow for a link to travel rectilinearly.

Example 1 – Sylvester–Kempe Inversor

Example Dimensions:
Cyan Links = $2$
Pink Links = $2$

Green Triangles:
Shorter Sides = ${\sqrt {2}}$
Longest Side = $2$

Yellow Triangles:
Shorter Sides = ${\sqrt {10}}$
Longest Side = $2{\sqrt {5}}$

Example 2 – Sylvester–Kempe Inversor

Example Dimensions:
Cyan Links = $3$
Pink Links = $3$

Green Triangles:
Shorter Sides = $2$
Longest Side = $2{\sqrt {2}}$

Yellow Triangles:
Shorter Sides = ${\sqrt {10}}$
Longest Side = $2{\sqrt {5}}$

Example 3 – Sylvester–Kempe Inversor

Example Dimensions:

Cyan Links = $4{\sqrt {10}}$
Pink Links = $4{\sqrt {10}}$

Green Triangles:
Shortest Side = $5$
Intermediate Side = $4{\sqrt {10}}$
Longest Side = $15$

Yellow Triangles:
Shortest Side = $5{\sqrt {5}}$
Intermediate Side = $20{\sqrt {2}}$
Longest Side = $15{\sqrt {5}}$

Example 4 – Kumara–Kampling Inversor

Created by Fumio Imai and Arglin Kampling. Rather than having the third joint of each triangular link be displaced off to the side, the third joint can also be displaced collinear to the original links, allowing for the links to remain as bars.

Example Dimensions:
Cyan Links = $1$
Pink Links = $1$
Green Links = $0.5{\sqrt {10}}+0.5{\sqrt {10}}$
Yellow Links = $0.5{\sqrt {2}}+0.5{\sqrt {2}}$

Notes

^ The midpoints must be displaced such that they not only form similar triangles, but also form a parallelogram (drawn in pink in the transition phase) if they are connected together. The triangles do not need to be right triangles, nor does the pink parallelogram have to be a rectangle. It is entirely coincidental that this happened.

References

^ Sylvester, J.J. (15 July 1875). "History of the Plagiograph". Nature. XII (298): 214–216. Bibcode:1875Natur..12..214S. doi:10.1038/012214b0.

External links

Quadruplanar Inversor Generalization – an interactive demo at GeoGebra for creating and simulating Quadruplanar Inversor linkages
A strong relationship between new and old inversion mechanisms Dijksman, E.A., Published in: Journal of Engineering for Industry : Transactions of the ASME, Published: 01/01/1971
https://americanhistory.si.edu/collections/search/object/nmah_1214012
https://alexandria.tue.nl/repository/freearticles/605221.pdf

[Note01-1] The midpoints must be displaced such that they not only form similar triangles, but also form a parallelogram (drawn in pink in the transition phase) if they are connected together. The triangles do not need to be right triangles, nor does the pink parallelogram have to be a rectangle. It is entirely coincidental that this happened.

[2] Sylvester, J.J. (15 July 1875). "History of the Plagiograph". Nature. XII (298): 214–216. Bibcode:1875Natur..12..214S. doi:10.1038/012214b0.

[Note 1]

[1]

@@ Line 1: / Line 1: @@
+{{Short description|Mechanism that provides a perfect straight line motion without sliding guides}}
 {{context|date=November 2016}}
+[[File:Quadruplanar Inversor Derivation Alt.gif|thumb|Animation to derive a Quadruplanar Inversor from [[Hart's inversors|Hart's first inversor]].<ref group=Note name=Note01/>
+]]
 The '''Quadruplanar inversor''' of Sylvester and Kempe is a generalization of [[Hart's inversor]]. Like Hart's inversor, is a mechanism that provides a perfect [[straight line motion]] without sliding guides.
 The mechanism was described in 1875 by [[James Joseph Sylvester]] in the journal [[Nature (journal)|Nature]].<ref>{{cite journal |first=J.J. |last=Sylvester |authorlink=James Joseph Sylvester |title=History of the Plagiograph |journal=[[Nature (journal)|Nature]] |volume=XII |issue=298 |pages=214–216 |date=15 July 1875 |doi=10.1038/012214b0 |bibcode=1875Natur..12..214S |url=http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=article&did=HistSciTech.Nature18750715.Sylve01&id=HistSciTech.Nature18750715|doi-access=free }}</ref>
-Like Hart's inversor, it is based on an [[antiparallelogram]] (KBCD in the figures below) but the rather than placing the fixed, input and output points on the sides (dividing them in fixed proportion so they are all similar), Sylvester recognized that the additional points could be displaced sideways off the sides, as long as they formed [[similar triangles]].  Hart's original form is simply the [[degenerate case]] of triangles with altitude zero.
+Like Hart's inversor, it is based on an [[antiparallelogram]] but the rather than placing the fixed, input and output points on the sides (dividing them in fixed proportion so they are all similar), Sylvester recognized that the additional points could be displaced sideways off the sides, as long as they formed [[similar triangles]].  Hart's original form is simply the [[degenerate case]] of triangles with altitude zero.
-== Gallery ==
+==Gallery==
 In these diagrams:
-* KBCD is an antiparallelogram, so KB = CD and BC = DK.
+* The antiparallelogram is highlighted in full opacity links.
-* KBA, CBB<sub>0</sub>, and CDE (and KDI in the first diagram) are similar triangles.
+* Yellow Triangles and Green Triangles are similar.
+** Green Triangles are congruent with each other.
-** Because KB = CD, KBA and CDE are congruent triangles
+** Yellow Triangles are congruent with each other.
-** Because BC = DK, CBB<sub>0</sub> and KDI are congruent triangles
+* Cyan links and Pink links are congruent.
-* A<sub>0</sub> and B<sub>0</sub> are fixed points.
+* Dashed links are additional appendages to allow for a link to travel [[Linear motion|rectilinearly]].
-* A<sub>0</sub>A = A<sub>0</sub>B<sub>0</sub>, i.e. A moves on a circle through B.
-* E moves in a straight line
+===Example 1 – Sylvester–Kempe Inversor===
-<gallery caption="Example 1">
-File:Quadruplanar invesor.gif|Animation
+[[File:Quadruplanar Inversor 1.gif|thumb|300px]]
+'''Example Dimensions:'''<br>
-File:Quadruplanar invesor.svg|Graphic
+Cyan Links = <math>2</math><br>
-File:Quadruplanar invesor description.svg|Description
+Pink Links = <math>2</math>
-</gallery>
-<gallery caption="Example 2">
+Green Triangles:<br>
-File:Quadruplanar invesor of Sylvester and Kempe.gif|Animation
+Shorter Sides = <math>\sqrt{2}</math><br>
-File:Quadruplanar invesor of Sylvester and Kempe.svg|Graphic
+Longest Side = <math>2</math>
-</gallery>
-<gallery caption="Example 3">
+Yellow Triangles:<br>
-File:Quadruplanar invesor of Sylvester and Kempe Alternate.gif|Animation
+Shorter Sides = <math>\sqrt{10}</math><br>
-File:Quadruplanar invesor of Sylvester and Kempe Alternate.svg|Graphic
+Longest Side = <math>2\sqrt{5}</math>
-File:Quadruplanar invesor of Sylvester and Kempe Alternate description.svg|Description
+{{clr}}
-</gallery>
-== References ==
+===Example 2 – Sylvester–Kempe Inversor===
+[[File:Quadruplanar Inversor 3.gif|thumb|300px]]
+'''Example Dimensions:'''<br>
+Cyan Links = <math>3</math><br>
+Pink Links = <math>3</math>
+Green Triangles:<br>
+Shorter Sides = <math>2</math><br>
+Longest Side = <math>2\sqrt{2}</math>
+Yellow Triangles:<br>
+Shorter Sides = <math>\sqrt{10}</math><br>
+Longest Side = <math>2\sqrt{5}</math>
+{{clr}}
+===Example 3 – Sylvester–Kempe Inversor===
+[[File:Quadruplanar Inversor 2.gif|thumb|300px]]
+'''Example Dimensions:'''
+Cyan Links = <math>4\sqrt{10}</math><br>
+Pink Links = <math>4\sqrt{10}</math>
+Green Triangles:<br>
+Shortest Side = <math>5</math><br>
+Intermediate Side = <math>4\sqrt{10}</math><br>
+Longest Side = <math>15</math>
+Yellow Triangles:<br>
+Shortest Side = <math>5\sqrt{5}</math><br>
+Intermediate Side = <math>20\sqrt{2}</math><br>
+Longest Side = <math>15\sqrt{5}</math>
+{{clr}}
+===Example 4 – Kumara–Kampling Inversor===
+[[File:Quadruplanar Inversor 4.gif|thumb|300px]]
+Created by Fumio Imai and Arglin Kampling. Rather than having the third joint of each triangular link be displaced off to the side, the third joint can also be displaced collinear to the original links, allowing for the links to remain as bars.
+'''Example Dimensions:'''<br>
+Cyan Links = <math>1</math><br>
+Pink Links = <math>1</math><br>
+Green Links = <math>0.5\sqrt{10} + 0.5\sqrt{10} </math><br>
+Yellow Links = <math>0.5\sqrt{2} + 0.5\sqrt{2}</math><br>
+{{clr}}
+==See also==
+<!--Alphabetical order, please! Thanks <3-->
+* [[Hart's inversors|Hart's first inversor / Hart's antiparallelogram / Hart's W-frame]], the origination of the Quadruplanar inversor.
+* [[Linkage (mechanical)]]
+* [[Straight line mechanism]]
+==Notes==
+[[File:Quadruplanar Inversor Derivation - Freeze Frame.png|thumb|A frame from the first animation for referencing to the note]]
+{{reflist|group=Note|refs=
+<ref name=Note01>The midpoints must be displaced such that they not only form similar triangles, but also form a parallelogram (drawn in pink in the transition phase) if they are connected together. The triangles do not need to be right triangles, nor does the pink parallelogram have to be a rectangle. It is entirely coincidental that this happened.</ref>
+}}
+==References==
 <references />
 ==External links==
-{{commonscat|Invesor of Sylvester and Kempe}}
+{{commons category|Invesor of Sylvester and Kempe}}
+* [https://www.geogebra.org/m/fbczmx5j Quadruplanar Inversor Generalization] – an interactive demo at [[GeoGebra]] for creating and simulating Quadruplanar Inversor linkages
 * [https://pure.tue.nl/ws/files/4439189/606433.pdf A strong relationship between new and old inversion mechanisms Dijksman, E.A.], Published in: Journal of Engineering for Industry : Transactions of the ASME, Published: 01/01/1971
-* http://americanhistory.si.edu/collections/search/object/nmah_1214012
+* https://americanhistory.si.edu/collections/search/object/nmah_1214012
-* http://alexandria.tue.nl/repository/freearticles/605221.pdf
+* https://alexandria.tue.nl/repository/freearticles/605221.pdf
 [[Category:Linkages (mechanical)]]
 [[Category:Linear motion]]
+[[Category:Straight line mechanisms]]
-{{engineering-stub}}