Realizing abstract simplicial complexes with specified edge lengths
Ellison, Matthew · arXiv (Cornell University) · 2023
For finite abstract simplicial complex $Σ$, initial realization $α$ in $\mathbb{E}^d$, and desired edge lengths $L$, we give practical sufficient conditions for the existence of a non-self-intersecting perturbation of $α$ realizing the lengths $L$. We provide software to verify these conditions by computer and optionally assist in the creation of an initial realization from abstract simplicial data. Applications include proving the existence of a planar embedding of a graph with specified edge lengths or proving the existence of polyhedra (or higher-dimensional polytopes) with specified edge lengths.