Quadruplanar inversor

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Animation to derive a Quadruplanar Inversor from Hart's first inversor. Quadruplanar Inversor Derivation Alt.gif
Animation to derive a Quadruplanar Inversor from Hart's first inversor.

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.

Contents

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.

In these diagrams:

Example 1 – Sylvester–Kempe Inversor

Quadruplanar Inversor 1.gif

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Green Triangles:
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Yellow Triangles:
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Example 2 – Sylvester–Kempe Inversor

Quadruplanar Inversor 3.gif

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Green Triangles:
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Yellow Triangles:
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Example 3 – Sylvester–Kempe Inversor

Quadruplanar Inversor 2.gif

Example Dimensions:

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Green Triangles:
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Yellow Triangles:
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Example 4 – Kumara–Kampling Inversor

Quadruplanar Inversor 4.gif

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.

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See also

Notes

A frame from the first animation for referencing to the note Quadruplanar Inversor Derivation - Freeze Frame.png
A frame from the first animation for referencing to the note
  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.

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References

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