What Are Mechanical Linkages?
Informally, a mechanical linkage is a system of rigid links (rods or bars) connected by ideal joints and moving in the plane or in the space. This definition suffices for engineering purposes, and one can find it in some form in many engineering books. However, from the mathematical viewpoint, this is not a satisfactory definition. Mathematically speaking, an abstract linkage is a finite connected metric graph $L=(G, \ell)$, a graph $G$ together with a length function $\ell$ which assigns to every edge $e\in E(G)$ of $G$ a positive real number, its length $\ell(e)$. Given such a graph and a target metric space $(X,d)$ (for the purpose of this article, $X$ will be the Euclidean space of some dimension), one defines the realization space $\mathcal{R}(L, X)$, as the space of maps from the vertex set of $G$ to the target space $X$, $f: V(G)\to X$, subject to the condition $$ d(f(v), f(w))=\ell([v,w]). $$ Here $v, w\in V(G)$ are vertices of $G$ and $[v,w]\in E(G)$ are the edges connecting these vertices. In terms of the informal definition of a mechanical linkage, each realization $f\in \mathcal{R}(L, X)$ defines a system of rods $f([v,w])$ connected at joints $f(v)$.
If $(X,d)$ is $\mathbb R^n$ with the Euclidean norm $||\cdot||$, then the length condition is better written as $$ ||f(v) - f(w)||^2= \ell^2([v,w]), $$ so that it becomes a polynomial equation on the vectors $f(v), f(w)\in \mathbb R^n$. With this in mind, the realization space becomes a real algebraic subset of $\mathbb R^{nm}$ (where $m$ is the number of vertices in $G$), given by the system of $k$ quadratic equations on $m$-tuples of vectors, %in $\mathbb R^n$: $$ ||\mathbf{x}_i - \mathbf{x}_j||^2=\ell^2([v_i, v_j]), $$ with vertices of $G$ enumerated as $v_1,...,v_m$. Here $k$ is the number of edges in $G$. The correspondence between $\mathcal{R}(L, \mathbb R^n)$ and this algebraic subset is given by $$ f\in \mathcal{R}(L, \mathbb R^n) \mapsto (f(v_1),...,f(v_m))= (\mathbf{x}_1,...,\mathbf{x}_m). $$ Incidentally, in a similar fashion one can get semialgebraic subsets of $\mathbb R^{nm}$, i.e. ones given by systems of polynomial equations and (nonstrict) inequalities. For instance, inequalities of the form $$ ||\mathbf{x}_i - \mathbf{x}_j||^2\le \ell^2([v_i, v_j]), $$ correspond to linkages where vertices $v_i, v_j$ are connected not by a rigid rod, but by a rope of length $\ell([v_i, v_j])$. But I digress. In fact, the system of equations defined above determines not just a subset of $\mathbb R^{nm}$, but a more complicated algebro-geometric object. Instead of defining it, I will consider the following simple example. Take $L$ to be the triangle (three vertices $u, v, w$ and three edges connecting these), with edge-lengths equal to $1=\ell([u,v]), 1=\ell([v,w])$ and $2=\ell([w,u])$. Then every realization of $L$ is a degenerate triangle, lying in a straight line. It is not hard to convince oneself that $\mathcal{R}(L, \mathbb R^2)$ is 3-dimensional, isomorphic (in any meaningful sense) to the product of the plane and the circle: A point in the plane corresponds to the location of $f(u)\in \mathbb R^2$ and a point in the circle corresponds to the length $2$ vector $\overrightarrow{f(u) f(w)}$. In total, one has three degrees of freedom for our linkage. However, something interesting happens in this example: If one linearizes the defining equations for $\mathcal{R}(L, \mathbb R^2)$, one gets not three but four degrees of freedom, one extra is given by the infinitesimal motion which keeps $f(u), f(w)$ fixed and moves $f(v)$ in a direction orthogonal to the vector $\overrightarrow{f(u) f(w)}$. An algebraic geometer will immediately recognize what is happening here (we have a nonreduced scheme), but I will keep the algebro-geometric language to a minimum. % and work with algebraic sets.
Some of the motions of a linkage in a space are more interesting than other: The ``uninteresting ones'' are obtained by modifying the given realization $f$ by postcomposition with a rigid Euclidean motion (one which preserves orientation). Considering realizations modulo rigid Euclidean motions corresponds to taking a suitable quotient of $\mathcal{R}(L, \mathbb R^n)$ by the group $SE(n)$ of orientation-preserving Euclidean isometries. In the case when the target space is the Euclidean plane, %with few trivial exceptions, taking such a quotient amounts to considering based realizations, ones where $f(v_1)$ is the origin and for an edge $e=[v_1,v_2]$ the image $f(v_2)$ has coordinates $(\ell(e), 0)$. The quotient $\mathcal{M}(L,\mathbb R^n)=\mathcal{R}(L, \mathbb R^n)/SE(n)$ is the moduli space of realizations of $L$ in $\mathbb R^n$. Since we assumed $G$ to be connected, the space of based realizations is a compact (and with few exceptions) algebraic subset of $\mathbb R^{n(m-1)}$.
As it turns out, in a sense, one can realize every compact real-algebraic subset $Y$ of $\mathbb R^N$ as the moduli space $\mathcal{M}(L,\mathbb R^2)$ of a planar linkage. Here I am lying a bit: One gets not $Y$ but several disjoint copies of $Y$, up to a Nash isomorphism: A polynomial map with algebraic inverse. This theorem has a long and complicated history, going back to the 18th century and involving steam engines, frogs, military secrets, Chebyshev polynomials, and work of A. B. Kempe (better known for the first incorrect solution of the 4-color problem). Ultimately, the theorem was proven in my paper with John Millson.
One can ask if a similar treatment, using graph theory, can be applied to more complex mechanical devices than just linkages (for instance, mechanical clocks, bicycles, analog computers, etc.). It turns out that the answer is again positive, but one has to consider more complex objects associated with metric graphs (vector bundles, connections and parallel sections), and instead of polynomial equations, one works with algebraic ordinary differential equations. But this is another story...