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Animation of the Cubic of Stationary Curvature of a Fourbar Linkage (goessner.net)
41 points by fango on July 25, 2017 | hide | past | favorite | 11 comments

I have no idea what I'm looking at, but it's beautifully illustrated. (And I wish it was interactive.)

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.

there are only three bars in the gif.

The base is considered the 4th bar.

I thought it might be something like that :).

If you liked that, check the videos by this Vietnamese Mechanical Engineer:


He is making/made Animator videos of almost any mechanism known ... https://www.youtube.com/user/thang010146/videos

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).


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

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


...and the Chebyshev "paradoxical" linkage:


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


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


...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?

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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