# Which curves and surfaces are realizable by linkages? references?

Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 below, but feel free to mention general d as well.

Let $$d\ge 1$$ be an integer, $$G=(V,E)$$ be a graph, $$w:E\to [0, \infty)$$ be a weight function. A realization in $$\mathbb R^d$$ with graph $$G$$ and weight $$w$$ is a map $$P:V\to\mathbb R^d$$ with the further property that $$|P(v)-P(v')|=w(v,v')$$ whenever $$\{v,v'\}\in E$$. I will identify such $$P$$ with its image, I hope it's not a problem.

Edit (May 11, 2020): As pointed out by Misha, this below definition is not correct. The action of isometries makes the set of all realizations of a linkage always cover all $$\mathbb R^d$$. He indicates a paper in which a more inclusive definition is formulated in $$d=2$$.

(Previous "wrong" definition: I say that a set $$A\subset \mathbb R^d$$ is realizable by linkages if there exists $$G,w$$ as above and an cover of $$A$$ by open sets of $$\mathbb R^d$$ such that for every $$U\subset\mathbb R^d$$ in the cover there exist $$G,w$$ such that that the union of all (images of) realizations of $$G,w$$ in $$\mathbb R^d$$ intersected with $$U$$ coincides with $$A\cap U$$.)

To "fix the problem", following the we will allow a subset of vertices of $$G$$ to be kept fixed in $$\mathbb R^d$$. In dimension $$2$$ this apparently generalizes the definition in the above paper, but I think that the result of the paper still allows to reply positively to the $$d=2$$ case of the question, with little extra work.

Revised definition: We say that $$A\subset \mathbb R^d$$ is realizable by linkages if there exists a cover of $$A$$ by open sets of $$\mathbb R^d$$ such that for every $$U\subset\mathbb R^d$$ in the cover there exist $$G=(V,E)$$ and $$w$$ as above, a subset $$F\subset V$$, and a map $$\phi: F\to\mathbb R^d$$, such that the union of all (images of) those realizations of $$G,w$$ which restricted to $$F$$ equal $$\phi$$, intersected with $$U$$, coincides with $$A\cap U$$.

Question: Say $$d=2$$ or $$d=3$$. Is it true that all algebraic sets $$A\subset\mathbb R^d$$ are realizable by linkages? What are references for this?

(Note: as of May 11 2020, it appears to me that the case $$d=2$$ is nicely treated in the answers given, while the case $$d=3$$ is not yet treated, possibly due to the previously bad definition.)

I found some mention of this, without references on Branko Grünbaum's "Lectures on lost mathematics", dated around 1975, and he says there that $$d=2$$ case is known, but does not give references, and $$d=3$$ case is a question by Hilbert which is open (but again no references there).

• Your definition of a realizable set is wrong: With this definition, the only realizable subsets are the open subset of ${\mathbb R}^d$. See here for precise definitions. – Misha May 8 at 18:58
• Thank you Misha, I will edit the question! – Mircea May 11 at 21:49

Erik Demaine and I also included a proof for $$d=2$$ in Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Chapter 3. There we asked if there is a planar (non-crossing) linkage that "signs your name" (traces any semi-algebraic region), a question posed by Don Shimamoto in 2004.
This was recently settled positively by Zachary Abel in his Ph.D. thesis: any polynomial curve $$f(x,y) = 0$$ can be traced by a non-crossing linkage.