
Animation of the Cubic of Stationary Curvature of a Fourbar Linkage - fango
http://s.goessner.net/#eulersavary
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pavel_lishin
I have no idea what I'm looking at, but it's beautifully illustrated. (And I
wish it was interactive.)

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kurthr
The four-bar linkage is a common design pattern in mechanical engineering.
With 4 rotating bearings and 4 bars, it allows one to constrain motion to a
plane. In particular, along a partucular path both position and angle smoothly
change together.

Once you start noticing them, you will see them everywhere (bus doors, clamps,
etc.). Their available paths are of course limited, but quite varied with
interesting tools for design.

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ouid
there are only three bars in the gif.

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prestonbriggs
The base is considered the 4th bar.

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ouid
I thought it might be something like that :).

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GregBuchholz
In reading the author's other article, "Cross Product Considered Harmful", and
I bet he would be interested in section 1.3 (starting on page 13) of the book
"An Introduction to Complex Analysis for Engineers" by Michael Alder. That
book introduces complex analysis using the matrix form of sqrt(-1).

[http://elibrary.bsu.az/azad/new/2554.pdf](http://elibrary.bsu.az/azad/new/2554.pdf)

~~~
fango
Thank you for the link. I was not aware of this. It's very interesting.

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GregBuchholz
Anyone have recommendations for learning about mechanical linkages
(books/MOOC/other)? Recently I've been intrigued by things like straight line
linkages:

[https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_lin...](https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage)

...and the Chebyshev "paradoxical" linkage:

[https://www.futilitycloset.com/2015/01/03/chebyshevs-
paradox...](https://www.futilitycloset.com/2015/01/03/chebyshevs-paradoxical-
mechanism/)

...and of course Kempe's "universality theorem", that there is a linkage that
traces any polynomial curve.

[https://arxiv.org/abs/1511.09002](https://arxiv.org/abs/1511.09002)

I recently came across "Planar Linkages Following a Prescribed Motion":

[https://arxiv.org/abs/1502.05623](https://arxiv.org/abs/1502.05623)

...which looks awesome. Especially intriguing are sentences like:

"In modern terms, the procedure proposed by Kempe is a parsing algorithm. It
takes the defining polynomial of a plane curve as input and realizes
arithmetic operations via certain elementary linkages. In this work, we
approach the question from a different perspective... ...By encoding motions
via polynomials over a noncommutative algebra, we reduce this task to a
factorization problem."

But currently the terminology used is considerably over my head. Anyone know
what branches of math you should study to be able to understand things like:

"...we recall that one can embed SE2 as an open subset of a real projective
space. This allows us to introduce a noncommutative algebra K whose
multiplication corresponds to the group operation in SE2, hence mimicking the
role played by dual quaternions with respect to SE3. A polynomial with
coefficients in K therefore describes a family of direct isometries, which we
call a rational motion."

...other suggestions for what to study to be able to synthesize new linkages?
Places or forums for a beginner to ask questions about linkages?

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theoh
It is really unusual to find "parsing algorithm" and "direct isometries" in
the same paper, so it's a bit of a genre-spanning publication. As the arxiv
tags say, the key fields could be described as "algebraic geometry" and "rings
and algebras". Things like SE2 refer to familiar groups (with useful geometric
interpretations) that mathematicians encounter frequently, so group theory is
a prerequisite also. Apart from the geometry, most of this is probably covered
by "abstract algebra" for which there are good resources online, including
lectures on YouTube.

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GregBuchholz
Thanks. I'll start looking into algebraic geometry and abstract algebra.

[https://www.amazon.com/Algebraic-Scientists-Engineers-
Mathem...](https://www.amazon.com/Algebraic-Scientists-Engineers-Mathematical-
Monographs/dp/0821815350/)

[https://www.amazon.com/Book-Abstract-Algebra-Second-
Mathemat...](https://www.amazon.com/Book-Abstract-Algebra-Second-
Mathematics/dp/0486474178/)

