2023/12/08 by Ellison, Matthew · 2 citations
#41-04 #51-04 #51M15 #51M20 #68U05 #Combinatorics (math.CO) #Computational Geometry (cs.CG) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.m #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2312.05376
For finite abstract simplicial complex Σ, initial realization α in 𝔼d, 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.