Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints

Abstract

We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset\mathbb R^d$ and $Q\subset\mathbb R^e$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack this question and verify it in three relevant special cases: if $P$ and $Q$ are centrally symmetric, if $Q$ is a slight perturbation of $P$, and if $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.

Figures (17)

• Figure 1: Non-isometric realizations with the same edge lenghts.
• Figure 2: A "pointed polytope", i.e., a polytope $P\subset\mathbb{R}^d$ with a point $x\in\mathop{\mathrm{int}}\nolimits(P)$. In addition to the edge lengths we also record the lengths of the gray bars -- the "vertex-point distances".
• Figure 3: Two non-isometric realizations of a pentagon with the same edge lengths and vertex-point distances. This is possible because the point is not in the interior.
• Figure 4: If $x\not\in\mathop{\mathrm{int}}\nolimits(Q)$, then it is possible for $Q$ to have shorter edges than $P$, while simultaneously also all vertices farther away from $x$.
• Figure 5: Two affinely equivalent (but not isometric) polytopes with the same edge length. The edge directions trace a circle on the "plane~at~infinity".

Theorems & Definitions (56)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture \ref{conj:main}
• Theorem 2.1
• proof
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Proposition 3.5
• proof