User:MWinter4/Maxwell-Cremona correspondence

The Maxwell-Cremona correspondence is a result in the intersection of graph theory, rigidity theory and polytope theory. It establishes a one-to-one correspondence between the equilibrium stresses, reciprocal diagrams and polyhedral liftings of a planar straight-line drawing of a 3-connected graph. Its most important consequence is that the planar drawing has a non-zero stress if and only if it has a non-trivial polyhedral lifting.

Relevant terminology
Let $$G=(V,E)$$ be a 3-connected planar graph. A planar straight-line drawing of $$G$$ is a map $$\boldsymbol p:V\to\Bbb R^2$$ that to each vertex $$v\in V$$ assigns a point $$p_v\in\Bbb R^2$$ in the plane, so that no two edges, drawn as straight lines between these points, intersect anywhere but their ends. The drawing divides the plane into polygonal regions. By $$F$$ we denote the set of these regions, or faces, of the drawing.

Equilibrium stresses
A stress for the drawing assigns to each edge $$e\in E$$ a real number $$\omega_e\in\Bbb R$$. The stress is said to be an equilibrium stress if


 * $$\sum_{w\,:\,vw\in E} \!\!\!\! \omega_{vw}(p_v-p_w) = 0\quad$$ for all vertices $$v\in V$$.

This has the following physical interpretation: each edge $$e=vw\in E$$ is considered as a spring with (potentially zero or negative) spring constant $$\omega_e$$ and equilibrium length zero. By Hook's law the spring pulls on its ends with force $$\pm\omega_{vw} (p_v-p_w)$$. In an equilibrium stress these forces cancel at each vertex.

Reciprocal diagrams
The dual graph $$G^*=(V^*,E^*)$$ has as vertex set $$V^*:=F$$ the regions of the drawing, two of which are adjacent if they bound a common edge. Each edge $$vw\in E$$ is incident to exactly two regions in the drawing, say $$f,g\in F$$, and hence corresponds to an edge $$fg\in E^*$$, and vice versa.

A straight-line drawing $$\boldsymbol q: F\to \Bbb R^2$$ of $$G^*$$ is called a reciprocal drawing or reciprocal diagram to $$\boldsymbol p$$ if corresponding edges $$vw\in E$$ and $$fg\in E^*$$ are drawn as lines that are perpendicular to each other. Formally,


 * $$(q_f-q_g)^\top(p_v-p_w)=0.$$

Polyhedral liftings
A lifting of $$\boldsymbol p$$ is a map $$\boldsymbol h:V\to\Bbb R$$ that to each vertex $$v\in V$$ assigns a height $$h_v$$. This yields a lifted drawing $$\bar{\boldsymbol p}:V\to\Bbb R^3$$ in 3-dimensional Euclidean space with $$\bar p_v:= (p_v,h_v)$$.

The lifting is a polyhedral lifting if the vertices that bound a common region in the darwing $$\boldsymbol p$$ are lifted to lie on a common plane in $$\Bbb R^3$$.

Formal statement
Let $$\boldsymbol p$$ be a planar straight-line drawing of a 3-connected planar graph $$G=(V,E)$$. Let $$F$$ be the set of regions in the drawing. Then there exists a one-to-one correspondences between the equilibrium stresses, reciprocal diagrams and polyhedral liftings of $$\boldsymbol p$$.

In particular, the following are equivalent:


 * $$\boldsymbol p$$ has a non-zero equilibrium stress $$\boldsymbol\omega: E\to\Bbb R$$.
 * $$\boldsymbol p$$ has a non-trivial reciprocal diagram $$\boldsymbol q:F\to\Bbb R^2$$.
 * $$\boldsymbol p$$ has a non-trivial polyhedral lifting $$\boldsymbol h:V\to\Bbb R$$.

The sign of the stress at an interior edge $$e\in E$$ determines whether the lifiting will be convex or concave at this edge: the stress at $$e$$ is positive/zero/negative if and only if the corresponding lifiting is convex/flat/concave at $$e$$. In particular, there exists a convex lifiting if and only if there exists a stress that is positive at all interior edges.

One-to-one correspondence
Let $$(G,\boldsymbol p)$$ be a planar straight-line drawing of $$G$$. Let $$ij\in E(G)$$ be an edge and $$uv\in E(G^*)$$ be the corresponding dual edge, appropriately oriented. It is a key ingredient to this proof that there is a way to choose globally compatible orientations of $$G$$ and $$G^*$$.

We will use the following notation:
 * $$(G,\boldsymbol q)$$ is the reciprocal diagram.
 * $$\boldsymbol \omega$$ is the stress.
 * $$h:V\to\Bbb R$$ is a lifting function, which to each vertex assignes a hight.

From a stress to a reciprocal diagram
Edges of the reciprocal diagram $$(G^*,\boldsymbol q)$$ are perpendicular to edges in $$(G,\boldsymbol p)$$. This is expressed by the following equation, in which $$R_{\pi/2}$$ is the 90°-rotation matrix:


 * $$q_u-q_v = \omega_{ij}R_{\pi/2}(p_i-p_j).$$

That this equation can be solved for $$\boldsymbol q$$ follows from the definition of the stress and the choice of a compatible orientation.

From a reciprocal diagram to a stress

 * $$\omega_{ij} := \frac{\|p_i-p_j\|}{\|q_u-q_v\|}.$$

From a polyhedral lifting to a reciprocal diagram
Let $$q_u$$ be the gradient of the face $$u$$.

From a reciprocal diagram to a polyhedral lifting
Solve


 * $$h_i-h_j = q_u^\top(p_i-p_j).$$

Tutte embeddings and Steinitz's theorem
Given any 3-connected planar graph $$G$$ and a non-separating induced cycle $$C$$ in $$G$$, the Tutte embedding yields a planar drawing of this graph in which $$C$$ is the outer face and where there exists a stress that is in equilibrium on every inner vertex. If the outer face is a triangle, then this stress can be extended to all edges.

This fact can be used to prove Steinitz's theorem if $$G$$ conatins a triangle. If it does not contain a triangle, then it contains a vertex of degree three, and one can instead construct a Tutte embedding of the dual graph. Lifting this embedding yields a realization of the dual polytope.

Weighted Delaunay triangulations
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